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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.

See also

• Product rule
• Quotient rule
• Chain rule
• Difference quotient