---
tag: 0QD4E5JYPT
kind: statement
role: proposition
name: Reciprocal rule
msc: [26A24]
uses: [0F1WR9C83K, 07ZMX4ANJS]
sources:
  - url: https://en.wikipedia.org/wiki/Reciprocal_rule
    locator: Proof
same_as:
  wikidata: Q1246553
  wikipedia: Reciprocal rule
---

## Statement

Let $g$ be a real function on an open interval $I$, differentiable at $a \in I$, with $g(a) \ne 0$.
Then $1/g$ is differentiable at $a$, and
$$\left( \frac{1}{g} \right)'(a) = -\frac{g'(a)}{g(a)^2}.$$

## Remarks

The difference quotient is $-\frac{g(a + h) - g(a)}{h} \cdot \frac{1}{g(a + h) g(a)}$. Continuity of $g$ at $a$ gives the limit.
For example, $(1/x)' = -1/x^2$ for $x \ne 0$.
Mathlib states it as `HasDerivAt.inv`.

## Explanations

### L1 human

When $g$ grows, $1/g$ shrinks, so the two derivatives have opposite signs.
The factor $1/g^2$ shows that $1/g$ changes fast where $g$ is small.