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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

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

where $g(x)\ne 0.$

Proof

From the quotient rule

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

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

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 $\cos x=0$) 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