What is Euler’s Method Formula in Calculus?

Learn about Euler’s Method formula and how it works along with some real-world examples to make it easy to grasp. Click here to find out more!

A common joke found all over the internet is a comment about teachers telling students who wouldn’t always have a calculator in their pocket. Obviously, the following comment is usually just an image of a modern cellphone. While we can all laugh at the joke behind this dialogue, the teacher is still right. A calculator on your phone can’t read your mind and write the equation for you.

That’s a broad statement, but it’s true for all forms of math from simple word problems to complex calculus of algebra problems. You need to know how to write out the equation and which equation to use. Understanding formulas like Euler’s Method are critical to solving many real-world problems and passing a college calculus class.

What Is Euler’s Method?

To understand Euler’s method, you’ll need to understand a few other math terms or formulas as well. The focus of this article is Euler’s method, so expect our definitions to be short and summarized. If you need more information on any of the topics, a quick search online will explain anything we leave out. That said, our definitions should convey all the knowledge you need to understand this method.

On occasion, you may need to solve a differential equation where you can’t use separation of variables, or you may get specific conditions to satisfy. Some of the methods you learn to conquer these types of equations simply won’t work. They should work, but the problem either ends up with an obviously incorrect answer and keep rolling over.

You can solve these types of differential equations using Euler’s method almost without failing. Why this isn’t the standard method taught in class is beyond us. That said, we’re going to help you understand Euler’s Method and how to use it. Before we get into the examples and a better explanation, let’s define some terms we’re going to use. The key terms we may use include:

  • Differential equations: This is an equation that uses a derivative to solve the equation for a function. 
  • Functions: This describes the relationship between the variables you put in and what comes out. It’s typically written out as f(x) or g(x). Each variable you put in has a relationship with the answer. That relationship is the function. Sine and tangent are trigonometry functions.
  • Derivatives: A derivative is usually used to define a slope or the change in a slope. You would use the slope formula to find the derivative of y = f(x). 
  • Separation of variables: This is typically how you solve differential equations by moving like terms to one side of an equation. The equation is generally noted by an equal sign. 
  • Tangent line: This is any line that comes in contact with a curve and mimics the curve where it runs into it. Small tangent lines are the basis of Euler’s Method. Tangent lines are usually outside of a curve if that curve is a circle.
  • Slope: This is any number that tells us the direction of a line and how steep that line may be along its path. A slope is equal to the rise divided by the run of the line.

Any other terms that require a definition will get defined in the same section. For now, those are the basic terms aside from understanding the basics of how to graph or work with fractions and variables. If you’re reading this, we assume you know how to create a graph or work with variables on some level whether it’s this advanced or on a lower tier.

Using tangent lines, Euler’s Method helps you approximate the solution to any equation, almost, if you know the initial value. If the problem changes rapidly or changes direction more than once, Euler’s Method may not work. That said, if the graph changes direction you’ll end up with multiple curves instead of one which rules out using Euler’s Method altogether. 

Euler’s Method is one of three favorite ways to solve differential equations. As we mentioned earlier, you may be able to use separation of variables, or you might find slope fields are the best method. Euler’s equation is must have a starting value or an assumed starting value in order to work. If you don’t have either of those things, refer to the other two methods we mentioned.

Using Euler’s method, we can see what goes on over a segment of our curve by intersecting or paralleling it with our tangent line. In short, Euler’s Method is used to see what goes on over a period of time or change. For instance, it can approximate the slope of a curve or define how money market funds changed over time. 

Using Euler’s Method, we can draw several tangent lines that meet a curve. Each line will match the curve in a different spot. By getting the approximate solution or equation where each line meets the curve, we can begin to put together a picture of what is happening along our curve. In short, Euler’s Method is just a lot of tangent lines strung together to help us guess at what the cure is doing as it travels.

Cons Of Using Euler’s Method

Since Euler’s Method only gives us approximate values, there may be room for error in the final result. If the curve is sharp and changes rapidly at any point, the solution we find at these sharp turns using Euler’s Method may lack accuracy. However, this is precisely where Euler’s excels if you need to crudely calculate why something sped up like rates of deaths due to disease or sales over a specified period.

While many people refer to Euler’s Method as a formula, and you can write a pseudo formula for it, it’s not a formula; it’s a method. It produces a solution without variables which may be considered an approximate value of the current problem. It also causes some issues with math teachers if they want you to use a specific formula or method.

When Would I Use Euler’s Method Outside Of Class?

Differential equations end up being a big part of our lives whether directly or indirectly. Mathematicians sometimes work with biologists to develop programs for monitoring diseases or population problems. Other mathematicians may work in banking or economics while some find their home in writing about things like Euler’s Method and exploring the different ways to solve differential equations.

Differential equations help scientists monitor everything from the Moon’s orbit to the rate at which a glacier may melt. We could keep giving examples, but we believe you understand how vital understanding this part of math is to your daily life and possibly your future career. Many professions beyond being a mathematician rely on approximations and solving differential equations.

You can check the Bureau of Labor Statistics for specific information on many job titles including those that use math a lot. Many careers at NASA require a firm grasp of applied mathematics and calculus. It’s how they figure out how to fly by a planet without hitting it which is probably essential. Pharmaceutical science uses calculus and is required for many jobs at places like Bayer as well.

Aerospace engineers spend their time creating and designing satellites, space vessels, space stations, and other human-made objects that need to survive in space. They rely on math heavily to do their jobs and ensure the safety of the equipment the build and the astronauts that us it.

Earthquake Safety Engineers rely on all forms of math to design better buildings and materials that withstand earthquakes. They also use math to build models to test their designs in a computer simulation. The same modeling steps may also be used to determine the damage existing structures might suffer during an earthquake

Aside from the complicated math used in the professions, we mentioned above, Euler’s method has many practical applications and may help determine simpler things like the rate of flow for running water. Similar methods and functions got used to help figure out how to shut off an oil leak that was 1,800 feet below the ocean’s waters. 

It’s impressive how math affects our lives every minute of the day. Many people don’t realize its importance, but we’re hoping we’ve conveyed the message well. That said, most common functions and formulas come preprogrammed into computers and calculators used in science-based fields. However, you still need to understand why and when to use them.

Some Final Notes

Euler’s Method is undoubtedly one of the most exciting formulas we’ve come across. Approximations usually find their home in less precise math problems. However, Euler’s method gets used across the spectrum of physics and various disciplines that use calculus. We don’t always need to know changes at every point along a curve, or we may have a starting value, and that’s when Euler’s Method works best.

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Slope Formula In Calculus Math: Our Comprehensive Guide

What is Slope Formula in calculus? To calculate the slope of a line you need only two points from that line, (x1, y1) and (x2, y2). Learn more here!

​The history of calculus and its many problems is almost as fascinating as its many modern applications and challenges. Gottfried Wilhelm Leibniz and Sir Isaac Newton discovered calculus in the 17th century. However, they did so independently then accused each other of stealing the other’s work. Their mathematical feud continued until the end of their respective lives.

Some historians argue that calculus was commonplace much earlier than the 17th century. Some claim it got used as early as 2000 BC. It’s true that some parts of calculus had been around for centuries before Newton’s and Leibniz’s discoveries, but they get credited with bringing all the functions, derivatives, integrals, and other terms into a group and calling it infinitesimal calculus. 

For clarity, calculus is here to help us solve problems, so we use the term problems as a way to describe an equation since it’s a problem until we answer it. The challenge in solving many problems is determining which formula applies and how to write the equation. Even a small error in a formula can wildly change the outcome when you’re dealing with math.

Photo of a board with formula in it

​Image via Pixabay

It sounds simple enough, right? However, if you knew a house cost X dollars in 1980 and it’s value increased until it was sold again in 1990 and you need to show at what rate the value of the house increased over time, how do you write that statement in mathematical terms? It sounds mind-numbingly hard, but the solution is simple if we use the right formula.

The calculus way to finding a solution to our problem is by creating a graph and finding the slope of the line that connects the original value of the house to its value at the time it got sold. The slope of the line indicates the rate at which the house increased in value. We’re assuming that the rate of increase created a straight line for this problem.

Defining the Slope Formula in Calculus

The slope of a line is always the same anywhere along the route. If we get technical, the slope is merely a ratio that defines the changes in a line as it goes from its starting point to its end. It’s easier to understand if you picture a graph and imagine the line starting at the bottom left corner of a graph and growing until it reaches any point along the right wall of the chart.

The changes in the line’s rise as it travels to its ending point on the right wall is the line’s slope. Let’s look at a real-world scenario like determining the slope of a roof. Picture a graph with a bottom row of evenly spaced numbers from zero to nine. The right wall of our graph has the same numbers on it except they go up vertically beginning with zero.

Our roof begins at the bottom left over the number zero and travels up until it ends at the number seven on the right wall of the graph. We know the graph is nine points across the bottom and the roof intersects the right wall at the number eight, we can write a simple formula to explain the slope which is: the slope equals eight divided by nine or 0.888 repeating. 

If the numbers on our graph indicate feet, we now know the roof rises about 0.88 feet per foot of roof length. This application of the slope formula has a lot of applications in the real world. For instance, an accessible wheelchair ramp at a public facility in some states must maintain a specific ratio. The ramp must be one foot long for every inch of rising. If the ramp is four inches tall, it must be four feet long. 

Our problem is simple to keep the math simple for the sake of explaining the slope formula. The math gets more complicated based on the type of slope. There are four types of slopes to contend with including:

  • Zero slope: the line is perfectly horizontal
  • Positive slope: this is when a line increases in height
  • Negative slope: this is a positive slope in reverse
  • Undefined slope: primarily used to define the slope of a vertical line because you can’t divide by zero

  • Real World Examples of the Importance of the Slope Formula

    We use math every day in hundreds of ways. Without math, we couldn’t drive cars or browse the internet. If you look closely, you’ll find some form of math in everything you do from cooking to exercising. Aside from built-in math that we all take advantage of, math is essential to many careers. You just can’t exist and not use math directly or indirectly.

    For most of us, we use math indirectly almost constantly. For instance, we know it takes 20 minutes to drive from home to work when traffic is slow. However, some days traffic is heavier than others so we use variables and math to determine when we should leave to ensure we make it to our job on time. Imagine the horror of baking without using math to measure ingredients.

    You may use math in a more direct sense if you work in medicine, construction, or marketing. If you do work in these fields, you know that the slope formula may be one of the most critical math formulas that we take for granted. It has different names depending on the occupation, but it’s all the same when you reduce it to pure math.

    The most apparent jobs that require a solid understanding of the slope formula are architects and construction workers. They use the slope formula or a variation of it when designing or building roofs, stairs, ramps, and just about every part of a building above ground. A typical house is little more than a box with a triangle nested on top of it. The slope formula is used to build the triangle on top. 

    The slope of the roof determines the number of materials needed to build it along with how to measure and cut each piece of material. It is much easier to construct a triangle out of wood and nails if you know how long each piece of wood needs to be to make the slope correct. In some cases, getting the slope wrong could cause the roof to collapse or perform poorly in high winds.

    Photo of different kinds of formula

    ​Image via Pixabay

    The slope formula makes everything around us, in respect to construction, safer. A slope is used to make roads safer and easier to travel across. It makes stairs and accessibility ramps safer and easier to use as well. Imagine building stairs without using the slope formula. Some might stretch out into the street while others may be too steep to climb. The same principle applies to roads and bridges.

    Social scientists and marketing professionals use the slope formula as well. Remember our example of a graph and determining the rise of a roof based on its run? The same principle gets used to determine things like how well a product is increasing concerning sales or how a population is performing based on specific economic indicators. 

    The slope of a line on a marketing graph tells a company how their product is performing. A negative slope indicates poor performance while a positive slope means the product is doing well. Determining the slope of the line on the marketing graph lets them know exactly how well the product is doing over time and may even indicate when the product’s sales are at their best.

    People that work in healthcare use the slope formula in a more critical sense. For example, epidemiologists may use a graph and the slope formula to determine how rapidly a disease is progressing over time. Doctors may use it to determine how long it might take for a treatment to cure a patient. The slope of a line may tell a doctor to increase or decrease medications as well.

    It’s critically important to healthcare to know the slope of lines on a chart if we expect doctors to find cures for terminal disease and many chronic ailments that don’t have treatments. It’s all about knowing how high a line rises over a specific period. That’s all math and 100 percent slope formula. 

    The list of uses for the slope formula is vast. If your career is impacted by it, you recognize its importance and understand why we think it’s one of the most crucial calculus topics. Beyond the examples we touched on above, the slope formula has many other uses including:

    • Determining speed over time
    • Determining distance over time
    • Calculating stock prices over time
    • Determining weight loss over time
    • Calculating win versus loss rates for sports teams
    • Calculating pay rate increases over time

    Some Final Notes

    Nothing we see or do today would be possible without math. Okay, a few things might be possible, but roads and stairs would be unsafe while some buildings might fall randomly. You wouldn’t be able to read this article without math because the internet and your computer could not exist without it. Appreciate the power of it and how the slope formula impacts your life at every angle.

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    Indeterminate Forms in Calculus: What are They?

    Indeterminate forms in calculus begin with algebraic functions that utilize a limit for the independent variable to find a solution. Learn more here!

    Image from Pixabay

    Indeterminate forms in calculus begin with algebraic functions that utilize a limit for the independent variable to find a solution. The indeterminate aspect of the form results when different rules vie to apply to the solution.

    Since there is no clear way to determine which rule of calculus will govern the answer, the function then becomes an intermediate form. There are many indeterminate forms, but there is only a handful that commonly occurs. For instance, 0/0 and 0 to the power of 0 are examples of more common indeterminate forms.

    What is an Indeterminate Form in Calculus?

    Replacing algebraic combinations within a function with its limits, assuming an independent variable sometimes results in an answer such as 0/0. In this case, there are several competing rules vying for an opportunity to define this solution.

    Indeterminate forms, officially coined by a student of the famed French mathematician Augustin Cauchy, have been around for as long as calculus. However, they have only been studied in the last 150 years or so. 

    Substituting a limit that results in a zero, infinity, negative infinity, or any combination of these may result in an indeterminate form. When both functions approach the given limit that results in the indeterminate form, there is not enough information to determine what the behavior of the function is at that point.  

    Some indeterminate forms can be solved by factoring through elimination or using L’Hopital’s rule. The functions resulting in 0/0 and infinity over negative infinity can achieve a solution through various means.

    An indeterminate form is a limit that is still easy to solve. It only means that in its current form as a limit put into a function, it presents too many unknowable characteristics to form an appropriate answer properly. You can’t just solve for the quotient.

    When solving for a limit, we are looking at two functions so that they make a ratio. Ratios are the most common, but not the only, way to discern an indeterminate function. Several geometric functions are also indeterminate forms, but not ratios.

    Finally, while limits resulting in zero, infinity, or negative infinity are often indeterminate forms, this is not always true. Infinity, negative or positive, over zero will always result in divergence. As well, one over zero has infinite solutions and is therefore not indeterminate. We can determine a universal solution, but an indeterminate answer is one that needs more information.

    Why are Indeterminate Forms in Calculus Important?

    Indeterminate forms hover over the calculus no matter where you turn. When dealing with ratios such as 1/0, 0/0, or infinity in any form, you will most likely need to use a further theorem, such as L’Hopital’s, to solve for a limit. Knowing that indeterminate forms are sometimes solvable can elicit clarity in a function that previously was not available.

    Delving further into the particulars of indeterminate forms allows us to utilize a variety of methods to determine the characteristics of a function at a particular limit. Not enough information determines an indeterminate form. However, sometimes, we have too much information and need to whittle down a solution to one technique.

    Take the fraction 0/0. An elementary view of this fraction would tell someone that it equals one because a numerator that equals a denominator equals one. Or more practically, we could think of the fraction as just zero. Further, a zero in the denominator could indicate infinity or does not exist. Therefore, there are options for classifying this ratio, but no clear winner stands out.

    So why are indeterminate forms necessary in practical, real-life situations? The answer is that they aren’t, at least not by themselves. However, their mere existence provides the ground for individuals to find ways to make indeterminate forms solvable.  

    Many areas of physics and math will use a denominator of zero as a potential starting point in a variety of situations and formulas. Laws involving physics, heat, and quantum mechanics will use indeterminate forms at some point. Understanding these forms allows you to solve them using an appropriate method, whether that is L’Hopital’s method or another.

    From an even more practical standpoint, there could be very general problems involving motion, velocity, and time that require the use of indeterminate forms. Questions surrounding the speed and time of a moving object will have to start at zero. Determining an infinite limit of two of these objects would result in an indeterminate form.

    Again, it is critical to note  the reason for classifying an indeterminate form as “indeterminate.” It is to differentiate it from other ratios that are zero or does not exist. To alleviate confusion, possibly substituting the word “temporary” for indeterminate would clear up some of the misconceptions surrounding the use of these forms.

    Understanding these forms as a transient is a better way to think of them. An indeterminate form, therefore, is just a vehicle for further computation that is up to the discretion of the user.

    How to Use Indeterminate Forms

    In the real-world, an indeterminate form would take the form of algebraic expression with a defined limit. Let’s say the numerator was x-4, and the denominator was 2x-8. With a limit of 8, what is the value of the ratio? The ratio becomes 0/0.

    Once again, 0/0 is an indeterminate form. Solving it further requires some factoring. If we cancel out the variable, we get a ratio of ⅔. Using simple algebra, we can get away from the indeterminate form. 

    When simple algebra is unavailable for use in solving an indeterminate form, then L’Hopital’s rule becomes necessary. A simple search of the internet results in some handy lists of indeterminate forms.

    Typically these lists are arrayed into three columns. The first column lists the indeterminate form. The second shows how to use L’Hopital’s rule to get a ratio to either 0/0 or infinity/infinity. Once the ratio becomes either of those, then they are solved either algebraically or using derivatives. 

    While L’Hopital’s rule is not precisely real-world applicable on its own, it is also a vehicle for further calculation much like indeterminate forms. Without the rule and the existence of indeterminate forms, calculations for meaningful and applicable formulas in other areas would not be possible.

    How do we know when an indeterminate form needs the use of L’Hopital’s rule or needs to use algebra?

    1. 1

      First, input the limit into the functions.

    2. 2

      Solve each function.

    3. 3

      If both the numerator and denominator equal zero or infinity, then you’ve probably got an indeterminate form.

    4. 4

      Attempt to factor functions without inputting the limit.

    5. 5

      If factoring results in a determinate form, then you are done. Otherwise, go on to the next step.

    6. 6

      Find the derivative of both the numerator and the denominator.

    7. 7

      Input your limit into the derivative ratio, and you’ve got your answer. You’ve just made use of L’Hopital’s rule.

    When using indeterminate forms, be sure you have a proper understanding of what a derivative is and how to use them. The first step in finding the derivative of a polynomial is to bring each exponent down by one. Multiply the initial exponent by the coefficient. Multiplying allows you to find the derivatives and use L’Hopital’s formula.

    It is critical to forming a thorough understanding of L’Hopital’s formula if you will be dealing with indeterminate forms. Instances where you could get end up with zero minus infinity, or infinity minus infinity all call for the use of this rule. The use of derivatives allows you to transform this indeterminate form into 0/0 or infinity over infinity. From there, you can solve using algebra. 

    There are instances where L’Hopital’s rule will not work with an indeterminate form. If you have zero over infinity, finding the derivative of the polynomials will only lead to the wrong answer. Using the inverse of the infinite polynomial will then change the ratio into an L’Hopital-friendly 0/0 ratio. 

    Instances, where a function equals zero to the zero power, requires the use of natural logarithms. Subtracting to infinities calls for using the laws of trigonometry and making calculations using cos, sin, and tan. Any indefinite forms that you find in the course of your calculus journey have a method for solving.


    While an indeterminate form may not change your life, it can provide the means for further understanding of broader calculus concepts and rules. As well, indeterminate forms are primarily made up of infinity, zero, and one, which is the primary values often dealt with in calculus. Understanding their indeterminate forms is crucial.

    Many subject areas will use formulas and calculations involving indeterminate forms. A thorough understanding of how to solve them, and the parts that make up their solutions such as derivatives and trigonometric functions, are fundamental.

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    Derivatives of Exponential Functions: Our Informative & Complete Guide

    Looking on how to calculate derivatives of exponential functions? Then you came into the right place! Here we listed everything you need to know about it.

    Once you’ve learned about derivatives and exponential functions separately, you can move on to calculating the derivatives of exponential functions. Let’s look at a few problems involving these equations, and break down the components you’ll need to know to solve them correctly.

    You’ll need to know about the following to solve these complicated problems:

    • Derivatives

    • Exponential functions

    • Variables

    • Transcendental numbers

    What is an Exponential Function?

    The definition of an exponential function can be explained by the following form:

    F(x) equals a x

    X is a variable, and the letter stands for the base of the function, or the mathematical term constant. The transcendental number e, which equals 2.71828, is the most used exponential function in calculus. When you use e, the expression turns into f(x) equal e x.   

    When you increase the exponent by one, the function value increases by the transcendental number e. When you decrease the exponent by one, the function value decreases by the same amount, that is, it’s divided by e.

    You will often encounter base-10 exponents in experimental science and electronics. The form for these problems is  f(x) equals 10x. Increase the exponent by one, and the base-10 function goes up by a factor of 10.

    Decrease the exponent, and the value decreases by one-tenth. A numerical change of one-tenths is referred to as one order of magnitude.

    When you have e, 10 or a similar constant base the exponential functional rescinds the logarithm, and the logarithm rescinds the exponential. Therefore, the logarithm and exponential functions are inverses of each other.

    An example: If you have a base of 10 and x equals three. Log (10 to the x power) equals log (10 to the third power). Then log equals 1000 equals three. Given that the base is 10 and the log equals 1000

    10(exponential log x) equals 10 (exponential log 1000) equals 10 to the third power equals 1000.

    What is a Derivative?

    A derivative is a tool used in differential calculus. Derivatives help determine rates of change, and slope lines of tangents. Derivatives have limits you need to use when working on equations.  

    Let’s look at the function f(x) equals 2x to the fifth power plus 7x cubed plus five.

    Use differentiation to determine the function that complements f. The second function is f’s derivative. Differentiation is the process you use to find the derivative.

    The derivative in this problem is f’ (x) equals 10 x to the 4th power plus 21x squared.

    A tick mark next to a variable indicates that it’s a derivative. Df means a change in the function (f), and dx means a change in x the variable. Df over dx is called a Leibnitz notation/ratio notation. The f with a tick mark is known as a prime or language notation.

    Derivatives can tell us about rates of change. D(t) can stand for the distance from your home in miles as a time function. It follows that D(2) equals five means you’re five miles to him after two hours of traveling.

    D’2 tells you two hours have passed. The formula is then the distance changes 20 miles for every hour that elapses equals 20 miles each hour equals 20 miles per hour. D’2(2) indicates that after two hours have gone by the velocity is 20 miles per hour.

    Another example: C'(x) is the cost of making x tons of spaghetti. Then C'(30) equals 15,000 lets us know that producing 30 tons of spaghetti costs $15,000. C’ (30) equals 48,000 indicates that 30 tons of spaghetti are being produced at the cost of $48,000 per ton.

    Example #1

    The expression used for the derivative of e to the x power is the same expression we used for e to the x power at the beginning of the problem. The problem reads d (e to the x power over dx equals d to the x power.

    The equation means that the slope equals the y-value, or function value, for all points on the graph. You can check a graph representation of the problem to verify this.

    Example #2

    Let’s look at a problem where x equals 2. At that point on the graph, the y value equals e to the second power, which is similar to or congruent to 7.39. The derivative of e to the x power is the same as e to the x power, so the tangent line slope located at x equals two is e to the second power, congruent or similar to 7.39

    Example #3

    Find the derivative of the exponential function y equals ten X to the third power.

    You’ll use this equation to get the answer:

    Dy over dx equals ten to the third power x (ln10) followed by d (3x) over dx equals three ln10 (10 to the third power x)

    Example #4

    Let’s look at a harder example. Find y equals sin (e to the third power x). Y will equal sin u if u equals e to the third power x.

    Dy over dx equals dy over du and du over dx, then (cos u) d (e to the third power x) over dx equals (cos e to the third power x)(3e to third power x), and then 3e to the third power x cos e to the third power x

    Example #5

    You’ll occasionally encounter derivatives of exponential functions that are even more complicated.

    Use the equation U equals cos2x and v equals e x squared to negative one. Follow it with du over dx equals 2 sin 2x, followed by dv over dx equals e to the x squared negative one power (2x), which results in the following equation:

    Dy over dx equals [cos 2x] [(x squared negative one (2x)] plus [ (e x squared negative one (negative 2 sin 2x] equals (e x squared negative 1) [cos2x(2x) minus 2sin2x]

    equals 2e x squared negative one[x cos2x minus sin2x]

    How to Understand Calculus, Algebra, and Math

    Calculus, algebra and any math get easier by practicing problems over and over again. Going to class and listening to lectures are a start, but they won’t help you get good grades or understand the material.  Memorizing problems won’t help you, either. Learn how to understand the why and how of what you’re doing.

    Calculus is difficult even for mathematically-inclined students, so you’ll need to go over points and formulas several times before you understand them. Don’t stick to just the problems you’ve been assigned. Look for other examples in your textbooks or online to help you learn derivatives of exponential functions and other calculus and algebraic functions.

    Read over the sections that you’re studying in class until you understand the process. Ask for help from your teacher or other students if you are having a hard time solving problems on your own. It is essential that you know why you are solving the problems.

    It helps to know the real-world applications associated with calculus problems, such as finding the slopes of tangent lines and rates of change.

    Person writing on a white notebook

    ​Image via Pexels

    Before you go to class, read over the previous week’s work (even if you think you know it well enough), and work on a few extra problems. A grasp of the last lesson will make it easy to understand new formulas and equations.

    Check your answers to practice questions, or better yet, have another student or a tutor check them for you. Work in a study team with one or more classmates. Explain the material you understand to them, and they’ll fill you in on formulas you don’t understand yet.

    Explaining calculus formulas to other students will help you learn more easily than merely listening to lectures and studying textbook examples by yourself. You may learn better from listening to other students to explain formulas than from listening to your teacher or another math expert.  

    Always have expectations for each study session, and try to meet or exceed them. Look for extra problems if you do better than expected. (There are lots of problems online of you run out of problems in your textbook.)

    Don’t get discouraged if you make mistakes during your study session. Take time to figure out where you made your error. Check online for help or ask a classmate or study partner. Learn the right way to solve a problem – like a derivative of an exponential function – before going on to the next mathematical concept.  

    Keep your textbook after the semester has ended. You may need to refer to during future studies. Old textbooks can serve as a refresher study tool after summer or winter break, along with your old class and study notes. Wait until you’ve finished all your math studies (or graduated) to dispose of your notes and sell your textbooks.

    Learn where to find the information you need to gain a better understanding of math formulas. People fail at math because they don’t adequately grasp the formulas needed for each type of problem and how and why to use that formula.

    Ask yourself (or the teacher) for examples of real-life reasons to use a formula, so it will be easier to apply yourself when practicing. You won’t simply be solving a bothersome math problem; you’ll be solving a scientific, architectural, or engineering problem.

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    What Is the Isosceles Triangle Theorem?


    What is the Isosceles Triangle Theorem, and why is it something you need to know? We answer this question and more. 

    Before we cover the Isosceles Triangle Theorem, we’ll discuss how we have used triangles over time in architecture, art, and design. Then we’ll talk about the history of isosceles triangles, the different types of triangles, and the different parts of isosceles triangles. 

    And lastly, after discussing the theorem, we’ll go over some useful formulas for calculating various parts of isosceles triangles.

    Various Applications of Triangles

    Isosceles triangles in a bed sheet design

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    While rectangles are more prevalent in architecture because they are easy to stack and organize, triangles provide more strength. With modern technology, triangles are easier to incorporate into building designs and are becoming more prevalent as a result. 

    In 1989, Japanese architects decided that a triangular building design would be necessary if they were to construct a 500-story building in Tokyo. The triangular shape could withstand earthquake forces, unlike a rectangular or square design.

    You can also see triangular building designs in Norway, the Flatiron Building in New York, public buildings and colleges, and modern home designs. ​​​​

    As far as isosceles triangles, you see them in architecture, from ancient to modern. Ancient Egyptians used them to create pyramids. Ancient Greeks used obtuse isosceles triangles as the shapes of gables and pediments. In the Middle Ages, architects used what is called the Egyptian isosceles triangle, or an acute isosceles triangle. 

    You can also see isosceles triangles in the work of artists and designers going back to the Neolithic era. They are visible on flags, heraldry, and in religious symbols. 

    The History of Isosceles Triangles

    The study of triangles is almost as old as civilization. Ancient Egyptians studied them as did Babylonians. You can see how ancient Egyptians used triangles to construct pyramids. 

    It wasn’t until about 300 BCE that the Greek mathematician, Euclid, gave triangles with two equal sides a name. He combined the Greek words “isos” (equal) and “skelos” (legs) to define them as triangles with exactly two sides. 

    Today, mathematicians define isosceles triangles as having at least two equal sides. This is a subtle but important difference because it means that equilateral triangles are also considered to be isosceles triangles. More on that below. 

    Other Types of Triangles

    In the world of geometry, there are many types of triangles besides isosceles:

    • Right triangles are triangles that have one right angle equaling 90 degrees. 

    • Scalene triangles are triangles with no equal sides. 

    • Acute triangles are triangles where all three angles are less than 90 degrees. 

    • Obtuse triangles have one angle that is greater than 90 degrees.

    • Equilateral triangles are triangles with three equal sides and angles.

    Isosceles triangles can also be acute, obtuse, or right, depending on their angle measurements. Equilateral triangles can be a type of isosceles triangle. However, note that not all isosceles triangles are equilateral. Moreover, an isosceles triangle can never be a scalene triangle. 

    There are a few particular types of isosceles triangles worth noting, such as the isosceles right triangle, or a 45-45-90 triangle. There is also the Calabi triangle, an obtuse isosceles triangle in which there are three different placements for the largest square. 

    And last but not least, there is also the golden triangle, which is an isosceles triangle where the duplicated leg is in the golden ratio to the distinct side. The golden ratio is defined as a ratio of two numbers in which the ratio of the sum to the bigger number is the same as the ratio of the larger number to the smaller. 

    The Parts of Isosceles Triangles

    Parts of an isosceles triangle

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    No matter how you define isosceles triangles, they are all made up of two legs and a base. If it’s an equilateral triangle, all sides can be considered the base because all sides are equal. 

    And, there are two equal angles opposite the equal sides. Each of these angles is called a base angle. The angle in between the legs is the vertex angle. 

    Also, the angles opposite each leg are equal and always less than 90 degrees (acute). All total, the angles should add up to 180 degrees. 

    The Isosceles Triangle Theorem

    Three triangles in color pink,yellow, and green

    Image via: Flickr

    Finally, it’s time to discuss the Isosceles Triangle Theorem. The Isosceles Triangle Theorem states: In a triangle, angles the opposite to the equal sides are equal. 

    So, how do we go about proving it true? It’s pretty simple. First, we’re going to need to label the different parts of an isosceles triangle. 

    Let’s give the points of the isosceles triangle the labels A, B, and D (counterclockwise from the top). We also need to draw a line from the center of the base (BD) to the angle (A) on the other side. Note that the center of the base is termed midpoint, and angles on the inside of the triangle are called interior angles. 

    Where that line intersects the side is labeled C. The line creates two triangles, ABC and ACD.

    So, Point C is on the base BD, creating line segment AC. Here’s what we have so far:

    • BC is congruent to CD (median)

    • AC is congruent to AC (reflexive property)

    • AB is congruent to AD (given)

    We have what is called the Side Side Postulate because all of the sides of ABC (three total) are congruent with ACD. If triangle ABC and triangle ACD have congruence, then their matching parts are congruent. This also proves that the B angle is congruent with the D angle.

    The Converse of the Isosceles Triangle Theorem

    reversed isosceles triangle

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    As with most mathematical theorems, there is a reverse of the Isosceles Triangle Theorem (usually referred to as the converse). It states, “if two angles of a triangle are congruent, the sides opposite to these angles are congruent.” Let’s work through it.

    First, we’ll need another isosceles triangle, EFH. The EFH angle is congruent with the EHF angle. We need to prove that EF is congruent with EH. To do that, draw a line from FEH (E is the apex angle) to the base FH. Label this point on the base as G. 

    By doing this, we have made two right triangles, EFG and EGH. Because we have an angle bisector with the line segment EG, FEG is congruent with HEG. So what is the result?

    We now have what’s known as the Angle Angle Side Theorem, or AAS Theorem, which states that two triangles are equal if two sides and the angle between them are equal.  

    Let’s take a look. We know that EFG is congruent with EHF. FEG is congruent with HEG. And EG is congruent with EG. That gives us two angles and a side, which is the AAS theorem. 

    When the triangles are proven to be congruent, the parts of the triangles are also congruent making EF congruent with EH. By working through everything above, we have proven true the converse (opposite) of the Isosceles Triangle Theorem. 

    Handy Calculations for Isosceles Triangles

    In addition to understanding the Isosceles Triangle Theorem, you should also be familiar with a few basic equations for isosceles triangles. 


    You can use the Pythagorean theorem to find the height of any triangle. First, label the two equal sides as a, and the base as b. The height (h) equals the square root of b2 – 1/4 a2. You can also divide the square root of 4a2 – b2in half and get the same result. 


    To measure the length of the outside of a triangle, add the length of each side together. But, since isosceles triangles have two equal sides, you can make the process easier with this formula: p = 2a + b.


    Area is defined as the total of unit squares you can fit inside any given shape. For an isosceles triangle, divide the total of the base (b) x height (h)by 2. 

    If you don’t know the height, use the formula listed above to calculate it.


    You can find the altitude of the isosceles triangle given the base (B) and the leg (L) by taking the square root of L2 – (B/2)2.


    To find the base of an isosceles triangle when you know the altitude (A) and leg (L), it is 2 x the square root of L2 – A2.  


    When given the base (B) and altitude (A), the leg is the square root of A2 + (B/2)2.

    Interior Angle

    When you know one interior angle of an isosceles triangle, it’s possible to find the other two. Let’s say that the angle at the apex is 40 degrees. Because angles must add up to 180 degrees, the two base angles need to add up to 140. Given they must be congruent angles, each of them must be 70 degrees. 


    Six different triangles

    Image via: Flickr

    In this article, we have covered the history of isosceles triangles, the different types of triangles, useful formulas, and various applications of isosceles triangles. 

    We also discussed the Isosceles Triangle Theorem to help you mathematically prove congruent isosceles triangles. For a little something extra, we also covered the converse of the Isosceles Triangle Theorem. You should be well prepared when it comes time to test your knowledge of isosceles triangles.  

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    What Is the Tangent Line Equation?

    In calculus you will come across a tangent line equation. What exactly is this equation? This article will explain everything you need to know about it.

    In calculus you will inevitably come across a tangent line equation. What exactly is this equation? This article will explain everything you need to know about it.

    In calculus, you learn that the slope of a curve is constantly changing when you move along a graph. This is the way it differentiates from a straight line. You can describe each point on a graph with a slope. 

    A tangent line is just a straight line with a slope that traverses right from that same and precise point on a graph. When we want to find the equation for the tangent, we need to deduce how to take the derivative of the source equation we are working with.

    When looking for the equation of a tangent line, you will need both a point and a slope. You will be able to identify the slope of the tangent line by deducing the value of the derivative at the place of tangency. This is where both line and point meet. 

    In regards to the related pursuit of the equation of the normal, the “normal” line is defined as a line which is perpendicular to the tangent. This line will be passing through the point of tangency. Now that we have briefly gone through what a tangent line equation is, we will take a look at the essential terms and formulas which you will need to be familiar with to find the tangent equation.

    Key Terms and Formulas Defined

    Circles, triangle and lines in geometry

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    Before we get to how to find the tangent line equation, we will go over the basic terms you will need to know. By having a clear understanding of these terms, you will be able to come to the correct answer in your search for the equation.

    • Tangent line – This is a straight line which is in contact with the function at a point and only at that specific point. This line is barely in contact with the function, but it does make contact and matches the curve’s slope. This line is also parallel at the point of the meeting. You can also simply call this a tangent.

    • Secant line – This is a line which is intersecting with the function. This line will be at the second point and intersects at two points on a curve. You can also just call this a secant.

    • Slope-intercept formula – This is the formula of y = mx + b, with m being the slope of a line and b being the y-intercept. You will use this formula for the line.

    • Point-slope formula – This is the formula of y – y1 = m (x-x1), which uses the point of a slope of a line, which is what x1, y1 refers to. The slope of the line is represented by m, which will get you the slope-intercept formula.

    With the key terms and formulas clearly understood, you are now ready to find the equation of the tangent line. You should retrace your steps and make sure you applied the formulas correctly. Otherwise, you will get a result which deviates from the correctly attributed equation.

    The Primary Method of Finding the Equation of the Tangent Line

    Magnifying glass turned in a paper

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    When we are ready to find the equation of the tangent line, we have to go through a few steps. If you take all these steps consecutively, you will find the result you are looking for. 

    There is more than one way to find the tangent line equation, which means that one method may prove easier for you than another. We will go over the multiple ways to find the equation. The following is the first method.

    Write Down the Function and Draw the Tangent Line

    While you can be brave and forgo using a graph to illustrate the tangent line, it will make your life easier to graph it so you can see it. This is because it makes it easier to follow along and identify if everything is done correctly on the path to finding the equation. You will want to draw the function on graph paper, with the tangent line going through a set point. 

    Use the Derivative For the Slope

    What you will want to do next is take the first derivative (f’x), which represents the slope of the tangent line somewhere, anywhere, on f(x), as long as it is on a point. 

    Input the x-coordinate Into f(x)

    Take the point you are using to find the equation and find what its x-coordinate is. When you input this coordinate into f'(x), you will get the slope of the tangent line.

    Convert the Tangent Line Equation Into Point-Slope Form

    What you need to do now is convert the equation of the tangent line into point-slope form. The conversion would look like this: y – y1 = m(x – x1). In this equation, m represents the slope whereas x1, y1 is a point on your line. Congratulations! You have found the tangent line equation.

    Confirm the Tangent Line Equation

    While you can be fairly certain that you have found the equation for the tangent line, you should still confirm you got the correct output. It helps to have a graphing calculator for this to make it easier for you, although you can use paper as well. You will graph the initial function, as well as the tangent line. If confirming manually, look at the graph you made earlier and see whether there are any mistakes.

    Methods to Solve Problems Related to the Tangent Line Equation

    Calculator with a text of problem solving

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    There are a few other methods worth going over because they relate to the tangent line equation. Knowing these will help you find the extreme points on the graph, the equation of the normal, and both the vertical and horizontal lines.

    Identify the Maximum and Minimum Points

    With this method, the first step you will take is locating where the extreme points are on the graph. These are the maximum and minimum points, given that one is higher than any other points, whereas another is lower than any points. Remember that a tangent line will always have a slope of zero at the maximum and minimum points. 

    A caveat to note is that just having a slope of 0 does not completely ensure the extreme points are the correct ones. To be confident that you found the extreme points, you should take the following steps:

    • Take the first derivative of the function, which will produce f'(x). The resulting equation will be for the tangent’s slope.

    • Solve for f'(x) = 0. This will uncover the likely maximum and minimum points.

    • Take the second derivative of the function, which will produce f”(x). What this will tell you is the speed at which the slope of the tangent is shifting.

    • For the likely maximum and minimum points that you uncovered previously, input the x-coordinate, a, into f”(x). Now you will have to check whether this is positive or negative. If it is positive, you have found the minimum at a. If it is negative, you have found the maximum. A third outcome you would get is the inflection point, when f”(a) equals zero. 

    • When you discover an extreme point at a, you will have to find f(a) to reveal the y-coordinate.

    Uncover the Equation of the Normal

    The “normal” to a curve at a specific point will go through that point. However, its slope is perpendicular to the tangent. When you want to find the equation of the normal, you will have to do the following:

    • Find the slope of the tangent line, which is represented as f'(x).

    • If you have the point at x = a, you will have to find the slope of the tangent at that same point.

    • You will now want to find the slope of the normal by calculating -1 / f'(a).

    • Write down the equation of the normal in the point-slope format.

    Vertical and Horizontal Tangent Lines

    To find out where a function has either a horizontal or vertical tangent, we will have to go through a few steps. 

    • When looking for a horizontal tangent line with a slope equating to zero, take the derivative of the function and set it as zero. 

    • Obtain and identify the x value.

    • Take the original function to deduce the y value. The result is that you now have the location of the point.

    • When looking for a vertical tangent line with an undefined slope, take the derivative of the function and set the denominator to zero. 

    • Obtain and identify the x value.

    • Take the original function to deduce the y value. The result is that you now have the location of the point.

    There are two things to stay mindful of when looking for vertical and horizontal tangent lines. In the case of horizontal tangents, you will want to make sure that the denominator is not zero at either the x or y points. In the case of vertical tangents, you will want to make sure that the numerator is not zero at either the x or y points.

    Wrapping Up

    Calculator with a text check

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    Congratulations on finding the equation of the tangent line! You can now be confident that you have the methodology to find the equation of a tangent. It may seem like a complex process, but it’s simple enough once you practice it a few times. The key is to understand the key terms and formulas. Having a graph as the visual representation of the slope and tangent line makes the process easier as well. 

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    Ultimate Guide On How To Calculate The Derivative Of Arccos

    Read this full guide on how to calculate the derivative of arccos that is used in trigonometry and understand the importance of degrees and radians.

    The derivative of arccos in trigonometry is an inverse function, and you can use numbers or symbols to find out the answer to a problem. It uses a simple formula that applies cos to each side of the equation. Some people find using a drawing of a triangle helps them figure out the solutions easier than using equations.

    The derivative of Arccos is used in trigonometry. It’s an inverse function, and you can manipulate it with numbers or symbols. There are several terms you’ll need to know when working with Arccos, including radian.

    Arccos means arccosine. You may also work with arcsin, or arcsine when working with trigonometry problems.

    Results for the derivative of Arccos are written in radians. Solving a problem involving Arccos first requires knowledge of radians.

    An Overview Of Radians

    Math symbols

    Image by ​Pixabay

    Radians are used to measure angles. The angle of a single radian subtends or creates an angle. At a certain point, extremities present straight lines that join at that particular point. When two rays pass through the arc endpoints, the angle is subtended.

    The radian has an angle of a single radian subtended from a unit circle’s center. A radian creates an arc length of 1. Therefore, we can determine that a full angle measures 2pi radians. The angle contains 360 degrees for every 2pi radians, which is equivalent to 57.29577951 degrees per radian or 180 degrees pi.

    A right angle measures pi/2 radians. A straight angle measures pi radians. Radians let you write integrals and derivative easily, like d/(dx)six equals cosx when x is measured by radians.

    The Difference Between Geometry And Trigonometry

    Math equations

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    The three branches of mathematics are algebra, arithmetic, and geometry. Geometry is the study of sizes, properties, and shapes of spaces of three- dimensional and two-dimensional objects. Euclid, the Father of Geometry, is often mentioned in conjunction with theorems and postulates about the field.

    Geometry is a combination of the Greek terms Geo (earth) and metron (measure). You will encounter solid, spherical and plane geometry in your studies.

    Spherical geometry is the study of three-dimensional objects such as spherical polygons and spherical triangles. Plane geometry features lessons about two-dimensional objects such as points, curves, lines, and planes, including polygons, circles, and triangles. Solid geometry focuses on spheres, prisms, cubes, pyramids, and other three dimensional objects.

    Euclidean Geometry is the study of flat surfaces, and it’s just one of the branches of geometry. Riemannian geometry, the study of curved surfaces, is another major branch of geometry.

    Trigonometry is a branch of geometry. The field emerged in about 150 B.C. when a mathematician named Hipparchus used sine to make a trig table.

    Trigonometry deals with triangles, their lengths, and their angles. You can also use trigonometry to study waves and oscillations. You will study the way the side lengths of right triangles relate to each other. The basic relationships between triangles and their sides are referred to as Sine, Tangent and Cosine.

    If you have a right angle triangle, the longest base is referred to as the hypotenuse. The side in front of an angle is its opposite side, and the adjacent side is the side behind that angle. The relationship for a right angle triangle then is sin A, or the opposite of the hypotenuse, cos A, the adjacent side of the hypotenuse, and tan A the opposite/adjacent side.

    The secondary relationships in trigonometry are:

    • Secant

    • Cotangent

    • Cosecant


    Image by Pixabay

    These measurements are the respective dimensions of Sine, Cosine, and Tangent.

    Spherical trigonometry that deals with triangles in 3D scenarios, and it’s used in navigation and astronomy.

    Trigonometry helped ancient sailors navigate the seas. Today, trigonometry has many uses beyond math class.

    Oceanographers use trigonometry to figure out the height of ocean tides or measure sea animals. Trigonometry can be used to build ships and help them navigate. This mathematical field also helps criminologists determine the trajectory of bullets and other projectiles.   

    Archeologists divide their excavation sites up for work using trigonometry. Flight engineers use trigonometry to determine the best flight course for a plane. Trigonometry helps fill in the “third side” of a flight’s wind, speed, and direction equation to make sure the plane goes in the right direction.  

    Example 1

    scuola student mathematical formula

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    In trigonometry, the Arccos means cos to the negative first power (z) or the arc cosine of the complex number represented by (z). You will consider that this problem involves an inverse trigonometry function and write down two identities:

    Cos (cos to the negative 1st power(x)) equals x, then cos to the negative 1st (power (cos(x)) equals x.

    You need to decide which formula to use. You should use the first formula, due to the chain rule. The chain rule allows you to find the derivative for the term cos−1

    (x) more easily than using the first formula. Simply the formula to cos (y) equals x. Now you have an implicit formula to solve the problem and determine y.

    Use implicit differentiation to find out the answer for y’.  

    (cos(y))0 = (x)followed by y’ (− sin(y)) = 1 and y’ equals negative 1 over sin (y), then y’ equals negative 1 over sin(cos to the negative 1st power(x)).

    The formula written above is an excellent tool, but there is an even better way to write it using geometry instead of algebra. Cos has an angle and a number between negative 1 and 1, cos negative one will also give angle and have a number between negative 1 and 1.

    Now define θ equals cos minus 1(x)to show the angle cos negative 1. Therefore, x equals cos (θ), according to the inverse function. You can also use cos with both sides of the function.

    Drawing a triangle can also help you find the answer to this problem. Draw a triangle with points ABC and θ will be the angle less than C. Cos (θ) equals ACBC in geometry, and cos (θ) also equals x.

    Consider that BC equals 1 and AC equals x. You could choose any identity in the above values, but the most obvious ones make the math easier.

    In the triangle example, sin(θ) = AB over BC. Use the Pythagorean Theorem to calculate the answer for sin(cos−1(x)), which is equal to sin(θ).

    Here’s the solution per the Pythagorean Theorem:

    BC squared equals AB squared plus AC squared, followed by AB squared equals BC squared minus AC squared. Therefore, AB squared equals one minus x squared, and AB equals the square root of 1 minus x squared.

    The equation calculated above answers the question correctly. Sin(cos negative 1 (x)) equals sin(θ) equals AB over BC equals AB equals the square root of 1 minus x squared. Complete the answer with

    Example #2

    Math symbols

    Image by Pixabay

    This problem demonstrates how to determine the derivative of Arccos x and Arcsin x.

    The equation you use is d over dx (arcsin x plus arccos x) equals zero. Note that arcsin x plus arccos x equals pi over 2. You can explain this equation with the following calculations:

    If arcsin x equals zero, then x equals sinθ equals cos( pi over 2 minus θ), then arccos x equals pi over 2 minus θ equals pi over 2 minus arcsin x; therefore arcsin x plus arccos x equals pi over 2.

    Example #3

    Scientific calculator

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    The derivative you find for arccos x will be the negative of arcsin x’s derivative. This derivative stature holds for the inverse of each cofunction pair.

    The same formulas apply to similar trigonometry problems. Arccot x’s derivative is the negative of arctan x’s derivative. Arcsec’s derivative is the negative of the derivative of arcsecs x.

    The variable y equals arcsec x, represent tan y equals plus-minus the square root of x to the second power minus one.

    Begin solving the problem by using y equals arcsec x, which shows sec y equals x. Now use Pythagorean identity b to figure the next step.  

    Tan y equals plus minus the square root of sec squared y minus 1 equals plus minus the square root of x squared minus 1. Y’s derivative equals arcsin x.

    The equation then is d over dx arcsin x equals one over the square root of one minus x squared. Prove this by looking at y equals arcsin x, which stands for sin y equals x. Figure the derivative of x with the following equation:

    Cos y followed by dy over dx equal 1, then dy over dx equals 1 over cos y’, then dy over dx equals 1 over the square root of 1 minus x squared ‘.

    An alternate theorem for this would be  d arcsin x over dx equals 1 over din sin y over dy equals 1 over cos y equals 1 over the square root of 1 minus x squared.

    Arccos X Functions

    The arccosine of x is the representation of the inverse cosine function of the variable x when negative 1 is less than x and x is less than 1. If the cosine of y equals x, the arccosine of the variable x is equal to x’s inverse cosine function, which equals y. The equation for this function reads arccos x equals cos to the inverse cosine x equals y.

    calculator and pen under the paper with solution

    Image Source: Pixabay

    The equation is written arccos 1 equals cos, followed by the negative 1 symbol, which stands for inverse cosine function in this problem.

    Therefore, arccos 1 equals cos negative 1 (inverse function), and 1 equals zero, and Rad equals zero degrees.

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    What is a Relative Minimum in Calculus?

    Learn about the relative minimum in calculus and how to find it along with when you might use the knowledge. Read further here!

    It’s easy to visualize the minimum and maximum values of a function since the minimum is the lowest point on a graph while the maximum is the highest. If you’re looking at a graph with points on it, just choose the lowest one. You can find four possible values of a function on a graph including:

    • Absolute maximum
    • Relative maximum
    • Absolute minimum
    • Relative minimum

    For the purpose of this article, we want to look at the relative minimum. The relative minimum won’t be the lowest point on the graph but a relative interval between the lowest and highest points. For instance, assume we have a graph with an x and y-axis. Let’s put three points five points on the chart that crisscross x as demonstrated.

    The first three points to the left of y end up being -9, 11, and -6. To the right of y, we have two points at 13 and -8. In this very simple scenario, the relative minimum is -6 and the relative maximum is 11 because they fall within the domain of x. The domain might get a little tricky at times, but it’s all simple if you think of it as merely being all the variables that make the function work.

    Understanding the Domain of a Function



    The domain of x could be a simple set of numbers beginning, from left to right, with negative six and ending with positive six. However, we don’t have to use the entire domain. Your function could end up confined between negative four and positive four. That’s how we end up with relative minimums and relative maximums. Other points may exist, obviously, but they don’t fall into the correct domain.

    Let’s look at a simple equation like y is equal to the square root of x plus four. The domain of this equation is illustrated with the function x is greater than or equal to negative four. We can’t go any lower than negative four of the equation breaks down because the denominator of a fraction can’t be a zero and number under our square root sign is positive and must remain that way.

    If you try to solve this using a number lower than negative four, it won’t work or produce an answer. Try it using -11 or even -5, and the result will always be incorrect. The domain of a function is a little more complicated than our simple scenario here, but to understand the relative minimum this definition is all you need at the moment.

    My Relative Minimum is Missing



    Sometimes on a test, you may have a problem that asked you to solve the equation x is equal to x squared using only two intervals like negative one and positive two. If you graph that, the relative and absolute minimum are both stuck on zero. The negative one can’t be the relative minimum because it’s at the end of our interval.

    In cases like this, the answer is always one of several multiple-choice solutions. The correct answer here would likely be none of the above. Passing a math test and math don’t always go together well, and little tricks like this get put on the test to make sure you’re paying attention not guessing the answer when you can’t solve the problem.

    Revisiting Relative Minimum

    If you didn’t notice, the first explanation of relative minimum is overly complicated and an almost robotic way of explaining it. You need to understand how to find the relative minimum using the most complicated methods possible if you want to excel at calculus. However, an easier way to find the relative maximum or minimum exists, if you graph everything.

    Assuming your graph has a line with a starting and ending point, and more than two points, the relative minimum is the lowest part of your line where it changes direction. The opposite is true for the relative maximum. If your line starts and negative eleven and moves to a positive number then back to negative six before climbing again, the negative six if where the direction changes and the relative minimum.

    Why Do I Need to Learn This?



    Well, for one, you need to know this to pass your algebra class. That seems like an excellent reason. Aside from that, many jobs require strong math skills and they pay well. If you check the Bureau of Labor Statistics website, you can look up your dream job and learn more about it. You can also use that website to look into other positions, find out what they pay, and possibly find a new career.

    However, many of the highest paying jobs require strong math skills. Understanding how to find the relative minimum or maximum is essential in several positions revolving around economics and marketing. However, the terms used in each industry may differ, and it’s unlikely your boss will ever email a spreadsheet and ask you to find the relative minimum of anything on it.

    You may be surprised to learn that some jobs you thought might be easy once you got the right degree actually require a high level of math knowledge. For instance:

    • Robotics engineers use a lot of math
    • Pilots that fly jets or commercial airliners must have a good grasp on math
    • Game designers need trigonometry to make things move and algebra to make the blow-up
    • Animators fall into the same category as game designers
    • Sports announcers use much more math than you can imagine

    Many countries require pilots to take numerical reasoning tests, and they don’t allow calculators. Take a practice test and see how well you do on it. While this isn’t a direct instance of you needing to derive the relative minimum of a function, it’s an excellent example of how vital understanding math might be to your academic and future careers.

    While they may call it by another name, several highly sought and well-paying jobs use calculus heavily. Let’s look at a couple of those with a little more detail starting with a mathematician job. They don’t just teach math, and many get employed at places like NASA. Check out NASA’s basic education requirements for some of their more important and fun jobs.

    A cryptographer is another excellent example of an almost a pure math-based job. They work everywhere these days from private companies that develop privacy products to governments that need to secure things for transmission. They also work at places like the NSA where they try to secure items and decipher elements that may be encrypted.

    Economists work with market data, statistics, and models to figure out current and future economic trends. Highs and lows on a graph might be their entire workload sometimes. Many of them work for various government agencies, but economist jobs in the private sector may become more popular in the future.

    People that work with investments like analysts and bankers need a lot of math to do their jobs well. Seeing highs and lows on a graph might be the only way they get to make a decision. Their decisions might cost someone a lot of money if they get the math wrong or don’t understand the graph.

    Accountants probably use math as often as a rocket designer. Math is practically all they do. They do things like figure out payroll for companies and prepare taxes for people. A mistake on their end could mean your paycheck is wrong or you end up paying the government back more than you owe. The opposite effects are possible as well.

    Geodesists tell us things like how far it is to celestial bodies like the sun or distant galaxies. Sometimes these objects move, and the distance may fluctuate which results in maximum and minimum distance measurements. They have to factor in many things mathematically like gravity and mass to get an accurate measurement.

    Mathematics modelers do several things including creating complex computer simulations and investigating number theories. If we need to know whether or not a comet will hit us, these people develop models based on data from various sources, including geodesists, to determine the chances of a comet smacking into the planet. That’s high on our list of essential jobs using math.

    Many of the higher paying versions of these jobs require a doctorate to compete in the job market.

    Some Final Notes

    Granted, out definitions and problems lean toward the simpler side of finding the relative minimum, but you should understand how it works from a broad viewpoint now. If you need more help, you can find plenty of videos online that explain how to arrive at the relative minimum and how it may apply to other pre-calculus and calculus problems.

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    Finding Limits In Calculus – Follow These Steps

    Finding limits in calculus isn’t easy. We’re here to help! Using these simple methods, we’ll be able to find the limit quickly no matter the situation.

    If you’ve been asked to find a limit by your teacher, there are a range of different methods that you can use. It’s much simpler than it sounds and at the end of this guide, we have a nifty strategy that you can follow so that you’ll always know which method to use and when.

    Finding limits isn’t easy, and a lot of people struggle with it. If this is you, don’t worry, by the end of this guide, we’ll have you finding limits in a few minutes at most. Read carefully and try to follow along, but don’t be afraid to start from the beginning to perfect your technique and to memorize each of the strategies properly.

    Why Would You Want to Find Limits?

    Limits are incredibly essential, and without them, we would be unable to do more advanced forms of calculus. A limit is the limit of a function f(x) as x approach c but never reaches it. Remember, x can approach c from either side. Picture a graph; it can come from either side of the axis.

    Limits allow us to find out how a function will behave even if it doesn’t exist at a specific value of x. The result is that finding limits will allow you to derive the angle of a slope at a given point, even if you don’t have a specific value of x for every point along the line. Without knowing how to find limits, we would have little information about the gradient between points.

    If we take the function f(x) = x – 1 / x – 1 and then imagine that x could be any number.

    We know that if x = 1 the function would look like this:

    f(1) = 1 – 1 / 1 – 1 which would equal 0 / 0.


    Image by Edmund Fung from Pixabay

    The result is that when x = 1, the function itself is undefined because the fraction 0 / 0 is undefined. On a graph, this would look like a straight line across, parallel to the x-axis, but there would be a gap where x = 1 because it’s just not defined. But what if we wanted to know what the function was when x = 1?

    Well, we can’t do it. But what we can do is to get as close as possible to x = 1 so that we can know approximately what the value of the function is at that point. This idea is necessarily a limit. It’s the idea of being able to get as close as possible to an undefined point so that we can approximate it with a high level of accuracy.

    Finding the Limit by Plugging in X

    The first technique that we’ll look at is plugging x into the function to see the limit. In an ideal world, this would work all of the time. Therefore, we always start with this technique because it’s the simplest and allows us to get more information about what to do next. The idea is that you make x equal to the number it ’s approaching.

    So, if we are trying to find the limit as we approach 2, we make x = 2 and then run the function.

    When you do this, you’ll get one of three results:

    • f(a) = b / 0 where b is not zero.

    • f(a) = b where b is a real number.

    • f(a) = 0 / 0.

    In the first circumstance, you’ve probably found an asymptote. An asymptote is when a line continually approaches a given value, but it will never reach it at any finite point.

    In the second situation, you have probably found the correct limit through the substitution method.

    Finally, in most complicated questions you will end up with a situation where the function is undefined, and therefore you’ll need to try other techniques. If this is the case, we will need to rearrange the function so that we can consider the limit in an identical but differently arranged form using one of the following three techniques.

    Factoring Method

    Factoring is a great method to try and is often one of the easiest to learn because it relies on skills that you’ve already practiced. If you’ve already tried to plug in a number have ended up with 0 / 0, you need to start factoring.

    Often you’ll see that either the numerator or the denominator is more ‘friendly’ to factoring. Usually, x with the highest power is the best place to start. Let’s consider the following equation:

    x^2 – 6x + 8 / x – 4 where x is approaching 4.

    In this example, the numerator is the only place for you to factor. It’s also obvious because of the x^2 which can factor. In this case, we can factor to:

    (x – 4)(x – 2) / (x – 4)

    As you can see, we can then cancel the two matching x – 4 on both the top and the bottom. Pretty simple, right? It won’t also be this easy, but if you continue to factor you can often find places to simplify the expression.

    This simplification leaves us with:

    f(x) = x – 2 where x is approaching 4.

    If we try to substitute 4 into the equation now, you’ll find the f(x) = 2. See, by factoring you’ve shown that the equivalent function has a specific value and that value is 2 when x is approaching 4.

    If you were to create a graph of this function, you would still see a gap where x = 4 because the original equation is still undefined. However, you’ll know that when approaching 4, the function equals 2.

    After factoring, you might find that there is no way for you to cancel and simplify. In this case, you should try another method to ensure that there is no limit of the function at the specific value of x.

    Rationalizing the Numerator

    The third technique requires you to rationalize the numerator so that you can try substitution again. You’ll know if you should rationalize the numerator because you’ll see a square root on the top and a polynomial expression on the bottom. Let’s look at the following example:

    f(x) = sqr(x-4) – 3 / x – 13 as the function approaches 13.

    We know that substitution fails when you get 0 in the denominator, and therefore substitution would fail in this example. Factoring would also fail because there is no polynomial to factor in this example.

    However, if you were to multiply the numerator and denominator by the conjugate of the top (numerator), then you’ll be able to cancel and find the limit.

    The conjugate of the numerator is: sqr(x – 4) + 3 and therefore we can multiply through to get:

    (sqr(x – 4) – 3)(sqr(x – 4) + 3) / (x – 13)(sqr(x – 4) + 3)

    We can then FOIL the numerator to get the following:

    (x – 4) + 3sqr(x – 4) – 3sqr(x – 4) – 9

    When simplified the above expression will become x – 13 because the middle terms cancel and then you can combine like terms.

    If we go back to the full equation you can now see that we have:

    (x – 13)  / (x – 13)(sqr(x – 4) + 3)

    The terms cancel, and we have:

    1 / (x – 13)(sqr(x – 4) + 3)

    From there, we can plug in 13 into the function because we have all of the unknowns on one side of the fraction. The result is that the limit is ⅙.

    Trig Identities

    So far we’ve only looked at situations which don’t include any trigonometry. These require unique methods like factoring and conjugates to ensure that you can simplify and be able to easily plug in a number for x to find the limit. We want to do the same with this equation, but it contains trigonometry which complicates things a little.

    For you to solve these equations, it’s vital that you know all of the trig functions so that you can rewrite equations and more effectively address them.

    The most common are as follows:

    Cos (x) = 1 / Sin (x)

    Sec (x) = 1 / Cos (x)

    Cot (x) = 1 / Tan (x)

    It’s highly likely that you’ll also need to know the double angle identities in order to simplify more complex functions.

    Sin (2a) = 2Sin(a)Cos(a)

    Cos (2a) = Cos^2(a) – Sin^2(a)

    Tan (2a) = 2Tan(a) / 1 – Tan^2(a)

    This equations might seem confusing, but they are actually very simple. They are each used for different purposes, but when finding limits we only need to know them for rewriting equations.

    Let’s look at the following example:

    Sin (x) / Sin (2x) when x is approaching 0

    We can use the double angle identities formula to simplify to:

    Sin (x) / 2Sin(x)Cos(x)

    From there, the Sin(x) can cancel and we are left with:

    1 / 2Cos(x)

    If we plug in 0 as x, we will get ½ because cos(0) = 1 and therefore you have 1 / 2*1 which is ½.

    The Strategy to Finding Limits in Calculus

    Now that we’ve covered all of the tactics that you can use to find limits let’s discuss which you should use and when. There is a straightforward rule. You should always do a direct substitution first.

    If you get f(a) = b / 0 then you have an asymptote.

    If you get f(a) = b then you have a limit.

    If you get f(a) = 0 / 0 then you should try factoring, rationalizing the numerator or trig identities depending on which seems most likely to work.

    Fetaured Image by Gerd Altmann from Pixabay

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    How To Calculate Hyperbolic Derivatives

    Hyperbolic derivatives may sound scary, but they are easy. In fact, their properties are among the most elegantly simple in all of mathematics.

    A hyperbolic derivative is a derivate of one of the hyperbolic functions, which are functions that utilize the exponential function (ex) to simplify otherwise complex calculations. Common uses for hyperbolic functions include representing the length of arcs such as those formed by the cables of a suspension bridge, or the shape of the Gateway Arch in St. Louis, Missouri.

    As derivatives calculate the rate of change of a function (such as deriving a function for a car’s acceleration given a function of its velocity over a variable of time), calculating a hyperbolic derivative means calculating the rate of change of a hyperbolic function. This definition is especially useful for understanding hyperbolic integrals, which are helpful in arc length calculations and other applications of mathematics and engineering.

    If some of these terms are unfamiliar (or outright intimidating), don’t worry! We’ll give a quick review of some of the basics before diving into the actual derivations.

    A Review ​Of Trigonometry

    Hyperbolic functions are closely related to trigonometry, as the functions that they describe relate to a hyperbola in the same way that trigonometric functions relate to a unit circle. But what exactly is this relationship, or even the “unit circle” for that matter? Is there a “unit hyperbola?”

    Trigonometry helps describe the “unit circle” represented by function x2+y2=1. On a two-dimensional graph, this circle has its center at the origin (point (0,0)) with a radius of one unit. Trigonometry is the study of the dimensions within this circle, where special functions provide the height and width of the circle’s radius at a certain angle. The “certain angle” part is especially important.

    This concept can be challenging to visualize internally, but consider an easy example: Suppose that the unit circle’s “certain angle” starts on the x-axis at zero degrees. In other words, if we position a line within the unit circle directly on the x-axis, it will be at zero degrees relative to the rest of the circle. Rotating it about the origin towards the positive y-axis will increase this angle in the positive direction while turning it towards the negative y-axis will decrease this angle in the negative direction.

    If we leave our imaginary line on the x-axis, what are the “components” of its height and width? Well, at zero degrees (or the x-axis), its width is its length. So, if our imaginary line has the same radius as the unit circle (one unit), then we could say that the “width” component of our circle is also one unit.

    However, what about our height component? Where do we start measuring? We measure the height component of our line from the x-axis, just as we measured its width from the y-axis. So, since our line is lying directly on the x-axis, its height component must be zero.

    As we rotate our imaginary line inside the unit circle, the height and width components of our imaginary line will change with the angle of rotation. Laying the line directly on the positive y-axis, for example, will have height and width components of one and zero units, respectively. But what about points in between?

    Thankfully, the height and width components of our line are conveniently described using basic trigonometric functions. Here, the height component is the function sin⁡(x) (pronounced “sine of x”), and the width component is the function cos⁡(x) (pronounced “cosine of x”). Using zero as a value for “x” in both of these equations will yield our original width and height of one and zero, respectively.

    We’ll come back to this subject later once we explore their similarities with the hyperbolic functions. Until then, we also have to review the all-important exponential function.

    A Review ​Of ​The Exponential Function (ex)

    The exponential function is one of the single most essential functions in all of mathematics. We won’t describe its full capabilities and applications here, but for now, understand that it adequately describes the growth of just about anything in the natural world: plant growth, radioactive decay and numerous other phenomena are all accurately quantified using the exponential function.

    The exponential function is represented by y=ex, where e=2.71828…. On a graph, this forms a hyperbola that grows at an exponential rate. e, also known as “Euler’s number,” just-so-happens to be a near-universal ratio of growth.

    This function is (hopefully) interesting and all, but how does it relate to hyperbolic functions?

    The Hyperbolic Functions: Combining Trigonometry ​And ​The Exponential Function


    Image Source : Pixabay

    Hyperbolic functions describe the hyperbola of the exponential function in a similar way that trigonometric functions describe the unit circle. The names of the basic “height” and “width” functions are remarkably similar to those from trigonometry!

    Where trigonometric functions include sin x and cos⁡(x), hyperbolic functions include sinh(x) (pronounced “cinch of x”) and koshx (pronounced “kosh of x” or “kosh” rhymes with “gosh”). These are defined using the exponential function such that:

    sinh x =ex-e-x2   and   x =ex+e-x2

    Note the subtle difference between the functions: The numerator of the sinh(x) function is ex-e-x, while that of the koshx function is ex+e-x.

    The names and functions of the hyperbolic functions aren’t their only similarities to trigonometry; other hyperbolic functions and identities are, in fact, remarkably similar to their counterparts in trigonometry.

    One example of this similarity is the “double angle” identity, which in trigonometry is sin 2x =2sin x cos x . What happens if we work backward and replace sin x and cos⁡(x) with their hyperbolic cousins? Something interesting happens:

    2sinh x x =2ex-e-x2ex+e-x2=e2x-e-2×2=sinh 2x

    The identity is the same as it is in trigonometry! Other trigonometric parallels are also possible but are somewhat beyond the scope of this guide.

    For now, we’ve defined the two most basic hyperbolic functions— sinh(x) and koshx. At this point, you’ve probably determined that finding a hyperbolic derivative involves finding the derivative of one of these (or one of the other) functions.

    Before actually calculating hyperbolic derivatives, however, we’ll have a quick review on regular derivatives. This review will be especially useful, as the exponential function has some very unique properties when it comes to derivation.

    A Review ​Of Derivatives

    Again, a derivative of a function is its rate of change with a variable. Returning to the car analogy, we might be able to describe a car’s distance traveled as a function of time. In other words, we could have a function that says our car moves a distance of “x” over time “t” such that x=50t.

    If we assign “x” to have its unit in meters and “t” to have its unit in seconds, we could say that our car will cover a distance of 50 meters in one second (which, by the way, is very fast—about 112 miles per hour!).

    We might be curious to know, then, how fast the car is going. Intuition tells us that if the car travels 50 meters in one second, it must have a speed of 50 meters per second. Just by going through this mental exercise, you’ve calculated a derivative!

    The exact theory of derivation is well beyond the scope of this guide. For now, remember that a derivative of a function describes that functions rate of change.

    When solving a derivative algebraically, one of the most powerful rules to remember is the “power rule.” Here, “power” refers to whatever power a function’s variable is raised to. For example, the function y=x2 has the variable “x” raised to a power of two.

    What would be the derivative of such a function? Or, in other words, what would be its rate of change as we change the value of “x?” The power rule offers a convenient solution, stating that the derivative of xn=n*xn-1. So, in the case of y=x2, dydx=2x (here, dydx means “the derivative of function ‘y’ by variable ‘x’”).

    What if we try that on our car’s position equation from earlier? A quick calculation shows that x’=1*50t1-1=50, which is exactly what our intuition told us.

    This rule works for most algebraic equations, but some functions have special derivation rules. One of these is the exponential function from earlier, whose derivative is – wait for it – itself! Formally stated, this relationship takes the form dydxenx=nenx. Take special care to remember any constant in front of the variable “x!”

    As you might be able to imagine, this relationship is especially helpful for calculating hyperbolic derivatives—which we can now finally do.

    Calculating Hyperbolic Derivatives


    Image Source : Pixabay

    Now that we have a solid background, we can move forward to deriving our essential hyperbolic functions. With the knowledge from the previous sections, the derivations should be reasonably straightforward.

    For calculating the derivative of sinh x , we derivate its value of ex-e-x2. Using the rules described in the previous section, this yields an exciting result: ex+e-x2. This value looks a lot like x because it is! Similarly, deriving x will produce the value of sinh x .


    It appears that the derivatives of the two essential hyperbolic functions sinh x and x are, in fact, each other. Remembering the parallels between hyperbolic and trigonometric identities, one can easily derive hyperbolic functions such as tanh x , where tanh x =sinh⁡(x)kosh(x) just as tan x =sin⁡(x)cos(x). Can you find its derivative?

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