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 $1/g(x)$ is given by

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

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,

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

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 $1/(x^2\; +\; 2x)$ 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 $cos\; xnot=0$) 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