# Reciprocal rule

In calculus, the reciprocal rule is a shorthand method of finding the derivative of a function that is the reciprocal of a differentiable function, without using the quotient rule or chain rule.

The reciprocal rule states that the derivative of $1/g(x)$ is given by

$\frac{d}{dx}\left(\frac{1}{g(x)}\right) = \frac{- g'(x)}{(g(x))^2}$

where $g(x) \neq 0.$

## Proof

### From the quotient rule

The reciprocal rule is derived from the quotient rule, with the numerator $f(x) = 1$. Then,

 $\frac{d}{dx}\left(\frac{1}{g(x)}\right) = \frac{d}{dx}\left(\frac{f(x)}{g(x)}\right)$ $= \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}$ $= \frac{0\cdot g(x) - 1\cdot g'(x)}{(g(x))^2}$ $= \frac{- g'(x)}{(g(x))^2}.$

### From the chain rule

It is also possible to derive the reciprocal rule from the chain rule, by a process very much like that of the derivation of the quotient rule. One thinks of $\frac{1}{g(x)}$ as being the function $\frac{1}{x}$ composed with the function $g(x)$. The result then follows by application of the chain rule.

## Examples

The derivative of $1/(x^3 + 4x)$ is:

$\frac{d}{dx}\left(\frac{1}{x^3 + 4x}\right) = \frac{-3x^2 - 4}{(x^3 + 4x)^2}.$

The derivative of $1/\cos(x)$ (when $\cos x\not=0$) is:

$\frac{d}{dx} \left(\frac{1}{\cos(x) }\right) = \frac{\sin(x)}{\cos^2(x)} = \frac{1}{\cos(x)} \frac{\sin(x)}{\cos(x)} = \sec(x)\tan(x).$

For more general examples, see the derivative article.