Definitions

# Reciprocal rule

This is about a method in calculus. For other uses of "reciprocal", see reciprocal.
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 $1/g\left(x\right)$ is given by

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

where $g\left(x\right) neq 0.$

## Proof

### From the quotient rule

The reciprocal rule is derived from the quotient rule, with the numerator $f\left(x\right) = 1$. Then,
$frac\left\{d\right\}\left\{dx\right\}left\left(frac\left\{1\right\}\left\{g\left(x\right)\right\}right\right) = frac\left\{d\right\}\left\{dx\right\}left\left(frac\left\{f\left(x\right)\right\}\left\{g\left(x\right)\right\}right\right)$ $= frac\left\{f\text{'}\left(x\right)g\left(x\right) - f\left(x\right)g\text{'}\left(x\right)\right\}\left\{\left(g\left(x\right)\right)^2\right\}$
$= frac\left\{0cdot g\left(x\right) - 1cdot g\text{'}\left(x\right)\right\}\left\{\left(g\left(x\right)\right)^2\right\}$
$= frac\left\{- g\text{'}\left(x\right)\right\}\left\{\left(g\left(x\right)\right)^2\right\}.$

### 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 $frac\left\{1\right\}\left\{g\left(x\right)\right\}$ as being the function $frac\left\{1\right\}\left\{x\right\}$ composed with the function $g\left(x\right)$. The result then follows by application of the chain rule.

## Examples

The derivative of $1/\left(x^2 + 2x\right)$ is:

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

The derivative of $1/cos\left(x\right)$ (when $cos xnot=0$) is:

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

For more general examples, see the derivative article.