Reciprocal rule

This is about a method in calculus. For other uses of "reciprocal", see reciprocal.

Topics in Calculus

Fundamental theorem Limits of functions Continuity Mean value theorem Differential calculus  Derivative Change of variables Implicit differentiation Taylor's theorem Related rates Rules and identities: Power rule, Product rule, Quotient rule, Chain rule Integral calculus  Integral Lists of integrals Improper integrals Integration by: parts, disks, cylindrical shells, substitution, trigonometric substitution, partial fractions, changing order Vector calculus  Gradient Divergence Curl Laplacian Gradient theorem Green's theorem Stokes' theorem Divergence theorem Multivariable calculus  Matrix calculus Partial derivative Multiple integral Line integral Surface integral Volume integral Jacobian

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 / (x² + 2x) is:

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

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