Reciprocal rule
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- 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
where
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[edit] Proof
[edit] From the quotient rule
The reciprocal rule is derived from the quotient rule, with the numerator f(x) = 1. Then,
[edit] 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 as being the function composed with the function g(x). The result then follows by application of the chain rule.
[edit] Examples
The derivative of 1 / (x2 + 2x) is:
The derivative of 1 / cos(x) (when ) is:
For more general examples, see the derivative article.