Reciprocal rule

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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'(x)}{(g(x))^2}

where g(x) \neq 0.

Proof[edit]

From the quotient rule[edit]

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[edit]

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[edit]

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\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[edit]