Reciprocal rule
From Wikipedia, the free encyclopedia
- 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
The reciprocal rule can also be derived from the chain rule. Let f(x) = x − 1. Then,
.
By 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.
