Derivative Rule for Inverse Functions

Definition

The derivative of an inverse function describes how the rate of change of a process is inverted when the roles of input and output are swapped. It relates the slope of $f^{-1}$ to the slope of $f$.

Why It Matters

Swapping inputs and outputs is a common operation in science and engineering. This rule allows us to calculate the rate of change for an inverted process without the complex task of finding a new algebraic formula for the inverse itself.

Core Concepts

• Reciprocal Rule: $(f^{-1})'(b) = \frac{1}{f'(a)}$, where $b = f(a)$.
  • How to read: “The derivative of f inverse at b equals one over f prime of a, where b equals f of a.”
  • Meaning: Inverse slopes are reciprocals at corresponding points.
• Functional Form: $(f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))}$.
  • How to read: “The derivative of f inverse at x equals one over the derivative f prime of f inverse of x.”
  • Meaning / when to use: Evaluate $f'(f^{-1}(x))$ then take reciprocal—avoids solving for $f^{-1}$ explicitly.
• Leibniz Notation: $\frac{dx}{dy} = \frac{1}{dy/dx}$.
  • How to read: “The derivative d x d y equals one over the derivative d y d x.”
  • Meaning: Swapping independent/dependent variables inverts the derivative.

Connected Concepts

• Derivatives of Inverse Trigonometric Functions
• One-to-One Functions
• Inverse Functions
• Derivatives of Exponential Functions