Differentiating Inverse Functions

What Differentiating an Inverse Function Means

Inverse functions reverse the input-output relationship of an original function. If , then the inverse function reverses that relationship so that .

For example, if , then its inverse is . Finding the inverse explicitly is sometimes easy, but it can be extremely difficult—or impossible to do conveniently—for more complicated functions.

The good news is that you can find the derivative of an inverse function directly from the original function. This is an important AP Calculus skill because many problems give you information about without ever asking you to find itself.

The Key Formula

If is differentiable and has an inverse, then the derivative of its inverse is:

This formula is worth understanding rather than simply memorizing. The derivative of the inverse at is the reciprocal of the derivative of the original function at the corresponding input.

Another equivalent form, which is often useful on the AP exam, is:

The second version says: if you know a point on the original function, then the corresponding point on the inverse is . The slopes at those two points are reciprocals.

Why the Formula Works

The relationship between a function and its inverse is:

Differentiate both sides with respect to . Because the inverse function is composed with the original function, the chain rule gives:

Solving for the derivative of the inverse gives:

So the inverse-function derivative rule is really a direct consequence of the chain rule.

A Step-by-Step Method

1. Identify the original function. Call it if useful.
2. Identify the point you need. Determine whether the problem gives you a point on or on .
3. Find the corresponding point on the other function. Remember that inverse functions swap the coordinates.
4. Find the derivative of the original function. Calculate at the appropriate input.
5. Take the reciprocal. Use the inverse-function derivative formula.
6. State the answer clearly. Be especially careful about which input value belongs in the derivative.

Worked Examples

The Most Important Idea: Swap the Coordinates

If the original function contains the point , then its inverse contains the point .

This is one of the easiest places to make an AP Calculus mistake. The input and output values switch when you move from a function to its inverse.

That means if a problem tells you and asks for the derivative of the inverse at , you should immediately recognize that the corresponding original input is .

Geometric Interpretation

The graph of an inverse function is the reflection of the original graph across the line . Reflection switches horizontal and vertical directions, so the slope of the inverse is the reciprocal of the original slope.

For example, if the original function has slope at a point, then the inverse has slope at the corresponding reflected point.

This geometric interpretation is a useful way to remember the formula: inverse functions flip the coordinates, so their slopes flip to reciprocals.

Common Mistakes

Practice Problems