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Practice Problem Set: Inverse Function Derivatives

A focused set of AP Calculus problems on derivatives of inverse functions, including numerical evaluation, inverse-function notation, and problems that require careful use of the inverse derivative formula.

The Key Formula

The derivative of an inverse function can be found without actually solving for the inverse function. If and the relevant derivative exists and is nonzero, then:

This formula is especially useful on AP Calculus questions where you are given information about but are asked to find the derivative of its inverse.

How to Use the Formula

  1. Identify the input to the inverse function. If you need , determine the corresponding value of .
  2. Use the inverse relationship. If , then .
  3. Find the original derivative. Evaluate at the corresponding original input.
  4. Take the reciprocal. The inverse derivative is the reciprocal of the original derivative.

Worked Examples

Example 1: Numerical Evaluation from a Table

Suppose and . Find .

Step 1: Since , the input to corresponds to the original input .
Step 2: Use the inverse derivative formula:
Step 3: Substitute the given derivative:
Result:
Example 2: Finding the Corresponding Original Input

Let . If , find .

Step 1: We need the original input such that .
Step 2: Solve :
Step 3: Differentiate:
Step 4: Evaluate at the corresponding input:
Result:
Example 3: Inverse Derivative from a Given Point

Suppose and . Find the slope of the inverse function at .

Step 1: The point on the original function becomes the point on the inverse.
Step 2: Apply the reciprocal rule:
Step 3: Substitute the derivative value:
Result:
Example 4: Chain Rule Connection

If and , explain why the inverse derivative is the reciprocal of the original derivative.

Step 1: Because inverse functions undo each other:
Step 2: Differentiate both sides using the chain rule:
Step 3: Solve for the derivative of the inverse:
Conclusion:

Common Mistakes

Taking the reciprocal of the wrong derivative

The denominator must be the derivative of the original function evaluated at the corresponding original input. Do not automatically plug the inverse-function input into without first identifying the matching original input.

Confusing inverse functions with reciprocal functions

is not the same thing as . An inverse function reverses the input-output relationship; a reciprocal simply divides 1 by a function's value.

Forgetting to switch the coordinates

If , then the inverse contains the point . This coordinate swap is often the key to locating the correct input for the original derivative.

Using a derivative of zero

The inverse derivative formula requires the original derivative to be nonzero at the relevant point. If , the reciprocal expression is undefined, so the usual inverse derivative formula cannot be used there.

Practice Problems

Question 1

Suppose and . Find .

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Question 2

Let . Find .

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Question 3

Suppose and . Find the slope of at .

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Question 4 (Find the Corresponding Input)

Let . If , find .

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Question 5 (Algebraic Inverse)

If , find using the inverse derivative formula rather than explicitly finding the inverse function.

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Question 6 (AP-Style)

A differentiable one-to-one function satisfies and . Find .

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