Reciprocal rule

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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

where g(x) ≠ 0.


From the quotient rule[edit]

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

From the chain rule and power rule[edit]

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.


The derivative of 1/(x3+4x) is:

The derivative of 1/cos(x) (when cos(x) ≠ 0) is:

For more general examples, see the derivative article.

See also[edit]