Reciprocal rule

This article is about a method in calculus. For other uses, 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(x) is given by

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

where g(x) ≠ 0.

Proof

From the quotient rule

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

{\begin{aligned}{\frac {{\mathrm d}}{{\mathrm d}x}}\left({\frac {1}{g(x)}}\right)={\frac {{\mathrm d}}{{\mathrm d}x}}\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}}}.\end{aligned}}

From the chain rule and power rule

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

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

Examples

The derivative of 1/(x³+4x) is:

{\frac {{\mathrm d}}{{\mathrm d}x}}\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 {{\mathrm d}}{{\mathrm d}x}}\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.

Reciprocal rule

Proof

From the quotient rule

From the chain rule and power rule

Examples

See also