Welcome to one of Albert.io’s AP® Exam Crash Course Reviews! These articles are an awesome way to get ahead on studying material from any of the available AP® exams, or to catch up on topics that you’ve fallen behind in. These guides take small topics that are often found on AP® exams and break them down into even smaller pieces, so that when you encounter them on the exam, you know exactly what to do!
In this review guide, we’re going to take a look at an AP® Calculus review topic: calculating derivatives of inverse function! Calculating the derivative of an inverse function requires you to apply derivation skills you’ve already learned to a specific type of function, inverse functions, which don’t always behave in a clear-cut manner. Have you worked with inverse functions before? If so, you’ll learn how to combine your knowledge of inverse functions with your knowledge of derivatives. If you haven’t, don’t worry! We will review what an inverse function is first, and then talk about how to take the derivative of one. In order to make sure you have the best possible understanding of how to calculate the derivative on an inverse function, we’re going to start with a step-by-step explanation containing plenty of examples, and then do a practice problem from an actual AP® Calculus exam! By the end of this AP® Calculus review article, you should be a pro at calculating the derivative of inverse functions. Are you ready? Let’s get started!
What is an Inverse Function?
To start, let’s define an inverse function. If we have an equation f\left( x \right)=x+1 , then the corresponding inverse function to this function is represented by f^{ -1 } \left( x \right) . Anytime you see a function with a ^{ -1 } in the superscript, you can assume that the function is an inverse function. In order to calculate an inverse function, you should set f(x) equal to x, and replace every instance of x within the formula with y. From there, you should solve the equation for y. Once you’ve solved the equation, replace y with f^{ -1 } \left( x \right) , and you have your inverse function! In the case of the example function we started with ( f\left( x \right)= x + 1 ), finding the inverse function would be done as follows:
Step 1: Replace every instance of x with y, and every instance of y with x.
To begin, take all of the x’s in your equation, and replace them with y’s, and vice versa. If your function begins with f(x) instead of a y, replace the f(x) with a y first, then change your variables.
f\left( x \right) = x + 1 x\rightarrow { } y = x + 1 x\rightarrow { } x = y + 1
Step 2: Solve the equation for y.
Next, solve the equation for y, as you normally would.
x = y + 1x\rightarrow { } x - 1 = y x\rightarrow { } y = x -1
Step 3: Replace y with f^{ -1 } \left( x \right)
Last step! Lastly, replace the y that you just solved for with f^{ -1 } \left( x \right) .
f^{ -1 } \left( x \right) = y = x - 1
And now you’re done! This same process can be used for any function that has an inverse.
In words, the inverse of a function ( f^{ -1 } \left( x \right) ) is the line that results from reflecting the function f(x) about the line y = x. If you want to make sure your inverse function has been calculated correctly, do so in two ways. First, you can add f^{ -1 } \left( x \right) and f(x) together. When added together, functions and their inverse functions always equal 1. Second, you can graph both lines to see whether they are reflections of each other.
How Do I Take the Derivative of an Inverse Function?
Now that we know what an inverse function is, we can learn to take the derivative of it! There are two main ways to approach differentiating an inverse function, and we will look at both.
The first way to approach differentiating an inverse function is to differentiate the inverse function directly. Continuing to use the example of f\left( x \right)= x + 1 , you would differentiate the inverse function by taking the derivative of it as follows:
f^{ ' }\left( f^{ -1 }\left( x \right) \right) = _{ dx }^{ d }{ f^{ -1 }\left( x \right) } = _{ dx }^{ d }{ (x - 1) = 1 }
The second way to approach taking the derivative of an inverse function is to create a formula that allows you to find the value of the derivative of the inverse function at any point using the original function that the inverse function is based on. Assuming that the function f(x) is differentiable (that is, it is a real-valued continuous function with no corners or cusps), the derivative of the inverse of f(x) can determined using the following formula:
{ (f }^{ -1 })'(x) = \dfrac { 1 }{ f^{ ' }\left( f^{ -1 }\left( x \right) \right) }
Now, this formula looks pretty complicated, so let’s look at an example of how to use it using our equation f\left( x \right)= x + 1 :
Step 1: Find the inverse of f(x) using the process outlines above
f\left( x \right) = x + 1 \rightarrow f^{ -1 }\left( x \right) =x - 1
Step 2: Find the first derivative of f(x)
_{ dx }^{ d }f(x)=_{ dx }^{ d }{ (x+1) }=1
Step 3: Substitute the inverse of f(x) for all instances of x in the first derivative
f'(f^{ -1 }\left( x \right) ) = f'(x - 1) = 0(x+1) + 1 = 1
Step 4: Find the reciprocal of Step 3
\dfrac { 1 }{ f'(f^{ -1 }\left( x \right) ) } = \dfrac { 1 }{ 1 } = 1 = _{ dx }^{ d }{ { (f }^{ -1 })'(x) }
And then you’re done! Although the equation may seem complicated, if you solve out the problem step by step, you’ll have no problem finding the derivative of an inverse function. Now, let’s look at some examples from the AP® Calculus exam!
AP® Calculus Exam Examples
On the 2007 AP® Calculus AP® exam free response section, one of the questions required you to take the derivative of an inverse function in order to complete it. Specifically, the question provided a table of values for a function g(x) and its first derivative g^{ -1 }\left( x \right) , and said that assuming g^{ -1 }\left( x \right) is the inverse of g(x), write the equation of the line tangent to the graph of g^{ -1 }\left( x \right) at x = 2. This problem also does not contain defined functions, so you must rely on your conceptual knowledge of inverse functions in order to solve it (in other words, the method of differentiation that we discussed earlier will not be useful for a problem like this). Let’s go over how this problem would be solved, step-by-step, using our knowledge of derivatives of inverse functions.
Step 1: Find the first derivative of g(x).
These values are given in the table provided, so we can come back to this once we know the inverse of g(x).
Step 2: Find the inverse of g(x).
This value is slightly trickier to find. We know that the value of g(x) at x = 1 is 2, and we know that the function g^{ -1 }\left( x \right) is the function g(x) reflected about the line y = x. The easiest way to determine the value of g^{ -1 }\left( x \right) then, is to swap the x and y values of g(1) = 2 to see that g^{ -1 }\left( 2 \right) . This method can be validated by plotting g(1) and y = x on a graph and determining what point mirrors g(1) = 2 on the other side of the line y = x. In this case, that value is 1 at x = 2, so g^{ -1 }\left( 2 \right) = 1 .
Step 3: Substitute the inverse of f(x) for all instances of x in the first derivative.
Knowing that g^{ -1 }\left( 2 \right) = 1 , we can now substitute g^{ -1 }\left( x \right) into { g }^{ \prime }\left( x \right) , which gives us the formula { g }^{ \prime }\left( 1 \right) . To determine the value of { g }^{ \prime }\left( 1 \right) , we can look at the table provided, which will show us that { g }^{ \prime }\left( 1 \right) =5.
Step 4: Find the reciprocal of Step 3
The last step to solving this portion of the problem is to take the reciprocal of the value determined in step three. This will give us the derivative of the inverse function g^{ -1 }\left( x \right) at x = 2, which, in this case, is equal to one-fifth.
From there, you can use this value as the slope of the tangent line for your equation, and use the point-slope formula to determine the equation of the tangent line to g^{ -1 }\left( x \right) at x = 2.
And there you have it! You’ve just solved part of a free response question that appeared on an actual AP® Calculus AB exam! Taking the derivative of an inverse function is a skill that is extremely important to understand, as it often appears in both the multiple choice and free response sections of the AP® Calculus exams. This AP® Calculus review is your first step to totally understanding how to approach questions involving derivatives of inverse functions on the AP® Calculus exam, but as you study, you should make a point of practicing questions that involve taking the derivative of inverse functions to make sure that you are prepared for any question that may appear on the exam.
Congratulations, you now have the tools to tackle any AP® Calculus exam question about derivatives of inverse functions! If you’ve just started studying for the AP® Calculus exam, check out some of our other review guides, so that you can set up the best possible plan to ace the AP® Calculus exam. Alternatively, if you’re interested in more AP® Calculus review articles that tackle smaller and more specific areas of the exam, check out some of our other Crash Course articles. We love feedback, so if you want to reach out and tell us about how you’re studying for your AP® exams, or which books you’re using, feel free to reach out! Otherwise, happy studying!
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