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Differentiating
Inverse Functions

Master the technique of finding derivatives of inverse functions using the Inverse Function Theorem and apply it to exponential, logarithmic, and trigonometric inverses.

AP Calculus AB

What Is an Inverse Function?

If \( f \) is a one-to-one function, then its inverse \( f^{-1} \) reverses the mapping: \( f^{-1}(f(x)) = x \) and \( f(f^{-1}(x)) = x \). Graphically, \( f^{-1} \) is the reflection of \( f \) across the line \( y = x \).

Visualizing \( f \) and \( f^{-1} \)

Why Differentiate Inverses?

Many important functions are defined as inverses: \( \ln x \) is the inverse of \( e^x \), and \( \arcsin x \) is the inverse of \( \sin x \). To find their derivatives, we need a systematic method rather than re-deriving each time.

\( \ln x \)
Inverse of \( e^x \). Its derivative \( \frac{1}{x} \) comes from the inverse rule.
\( \arcsin x \)
Inverse of \( \sin x \). Derivative \( \frac{1}{\sqrt{1-x^2}} \) follows from the theorem.
\( \arctan x \)
Inverse of \( \tan x \). Derivative \( \frac{1}{1+x^2} \) uses the same technique.

The Inverse Function Theorem

In words: the derivative of the inverse at a point equals the reciprocal of the derivative of the original function, evaluated at the corresponding point.

Geometric Intuition

Proof via Implicit Differentiation

1
Start with the identity: \( f(f^{-1}(x)) = x \)
2
Differentiate both sides: \( f'(f^{-1}(x)) \cdot (f^{-1})'(x) = 1 \)
3
Solve for the inverse derivative: \( (f^{-1})'(x) = \dfrac{1}{f'(f^{-1}(x))} \)

\( f^{-1}(x) \) vs. \( \frac{1}{f(x)} \)

\( f^{-1}(x) \) \( \frac{1}{f(x)} \)
Meaning Inverse function (undoes \( f \)) Reciprocal of output value
Notation \( f^{-1}(x) \) \( [f(x)]^{-1} \)
Derivative \( \dfrac{1}{f'(f^{-1}(x))} \) \( -\dfrac{f'(x)}{[f(x)]^2} \)
Example \( \sin^{-1}(x) = \arcsin(x) \) \( \frac{1}{\sin(x)} = \csc(x) \)

How to Apply the Theorem

1
Identify the function \( f \) and the point where you need \( (f^{-1})'(x) \).
2
Find \( f^{-1}(x) \): Determine the value \( a \) such that \( f(a) = x \). Then \( f^{-1}(x) = a \).
3
Compute \( f'(a) \): Differentiate \( f \) and evaluate at \( a = f^{-1}(x) \).
4
Take the reciprocal: \( (f^{-1})'(x) = \dfrac{1}{f'(a)} \).

Example 1: Polynomial Inverse

Let \( f(x) = x^3 + x \). Find \( (f^{-1})'(2) \).

1
Find \( a \) where \( f(a) = 2 \): Solve \( a^3 + a = 2 \). By inspection, \( a = 1 \) works since \( 1^3 + 1 = 2 \).
2
Find \( f'(x) \): \( f'(x) = 3x^2 + 1 \). So \( f'(1) = 3(1)^2 + 1 = 4 \).
3
Apply the formula: \( (f^{-1})'(2) = \dfrac{1}{f'(1)} = \dfrac{1}{4} \).

Example 2: Deriving \( \frac{d}{dx}[\ln x] \)

Since \( \ln x = f^{-1}(x) \) where \( f(x) = e^x \), we can derive its derivative using the inverse function theorem.

1
Identify: \( f(x) = e^x \), so \( f^{-1}(x) = \ln x \).
2
Compute \( f'(x) \): \( f'(x) = e^x \).
3
Apply theorem: \( \frac{d}{dx}[\ln x] = \dfrac{1}{f'(f^{-1}(x))} = \dfrac{1}{e^{\ln x}} = \dfrac{1}{x} \).

Example 3: Deriving \( \frac{d}{dx}[\arcsin x] \)

Let \( y = \arcsin x \), so \( \sin y = x \). We differentiate implicitly.

1
Differentiate: \( \cos y \cdot \dfrac{dy}{dx} = 1 \), so \( \dfrac{dy}{dx} = \dfrac{1}{\cos y} \).
2
Express in terms of \( x \): Since \( \sin y = x \), we get \( \cos y = \sqrt{1 - \sin^2 y} = \sqrt{1 - x^2} \).
3
Result: \( \dfrac{d}{dx}[\arcsin x] = \dfrac{1}{\sqrt{1 - x^2}} \).

Inverse Trig Derivative Table

Function Derivative Domain
\( \arcsin x \) \( \dfrac{1}{\sqrt{1 - x^2}} \) \( -1 < x < 1 \)
\( \arccos x \) \( -\dfrac{1}{\sqrt{1 - x^2}} \) \( -1 < x < 1 \)
\( \arctan x \) \( \dfrac{1}{1 + x^2} \) \( (-\infty, \infty) \)

Example 4: Using a Table of Values

Given the table below for a one-to-one function \( g \), find \( (g^{-1})'(7) \).

\( x \) \( g(x) \) \( g'(x) \)
\( 1 \)\( 4 \)\( 3 \)
\( 2 \)\( 7 \)\( 5 \)
\( 3 \)\( 11 \)\( 8 \)

Common Mistakes

✕ Wrong evaluation point
Evaluating \( f' \) at \( x \) instead of at \( f^{-1}(x) \). Remember: plug in \( f^{-1}(x) \), not \( x \).
✕ Confusing inverse and reciprocal
Writing \( (f^{-1})'(x) = \frac{1}{f'(x)} \) instead of \( \frac{1}{f'(f^{-1}(x))} \). The argument matters.
✕ Forgetting the sign for \( \arccos \)
The derivative of \( \arccos x \) has a negative sign. Missing the negative is a frequent AP error.

Summary Table

Technique Formula When to Use
Inverse Function Theorem \( (f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))} \) When \( f^{-1} \) is not explicit
Implicit Differentiation Differentiate \( f(y) = x \) Deriving inverse trig formulas
Table Lookup Match \( f(a) = x \), use \( 1/f'(a) \) AP free-response with data tables

Question 1 of 5

If \( f(x) = x^5 + 3x + 1 \) and \( f(1) = 5 \), what is \( (f^{-1})'(5) \)?

Question 2 of 5

What is \( \frac{d}{dx}[\arctan x] \)?

Question 3 of 5

If \( h(3) = 10 \) and \( h'(3) = -2 \), what is \( (h^{-1})'(10) \)?

Question 4 of 5

The derivative \( \frac{d}{dx}[\arcsin(2x)] \) equals:

Question 5 of 5

If \( f(x) = e^{2x} + x \), which expression equals \( (f^{-1})'(f(0)) \)?