---
tag: 0M0RZ3HSB0
kind: statement
role: theorem
name: Derivative of an inverse function
aliases: [inverse-function-rule]
msc: [26A24]
uses: [07BY47HG5S, 0F1WR9C83K, 0RXX2PT1MQ]
sources:
  - url: https://en.wikipedia.org/wiki/Inverse_function_rule
    locator: Derivation
same_as:
  wikidata: Q2143416
  wikipedia: Inverse function rule
---

## Statement

Let $I$ be an open interval and let $f \colon I \to \mathbb{R}$ be continuous and strictly monotone.
Assume that $f$ is differentiable at $a \in I$ with $f'(a) \ne 0$, and let $b = f(a)$.
Then $f^{-1}$ is differentiable at $b$, and
$$(f^{-1})'(b) = \frac{1}{f'(a)} = \frac{1}{f'(f^{-1}(b))}.$$

## Remarks

The inverse is [[continuous|07BY47HG5S]], so $f^{-1}(b + k) \to a$ as $k \to 0$. The difference quotient of $f^{-1}$ at $b$ is the reciprocal of a difference quotient of $f$ at $a$.
If $f'(a) = 0$, then $f^{-1}$ is not differentiable at $b$. An example is $f(x) = x^3$ at $a = 0$.
For example, it gives the [[derivative of the logarithm|07X1QA6BRW]] from the derivative of the exponential function.

## Explanations

### L1 human

The graph of $f^{-1}$ is the graph of $f$ reflected in the line $y = x$.
The reflection turns a slope $m$ into the slope $1/m$.