Reciprocal rule

:"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

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

where $g(x)\; eq\; 0.$

Proof

From the quotient rule

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

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:

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

The derivative of $1/cos(x)$ (when $cos\; x\; ot=0$) is:

:$frac\{d\}\{dx\}\; left(frac\{1\}\{cos(x)\; \}\; ight)\; =\; frac\{sin(x)\}\{cos^2(x)\}\; =\; frac\{1\}\{cos(x)\}\; frac\{sin(x)\}\{cos(x)\}\; =\; sec(x)\; an(x).$

For more general examples, see the derivative article.

ee also

*Product rule
*Quotient rule
*Chain rule
*Difference quotient