The reciprocal rule states that the derivative of 1/g(x) is given by
where g(x) ≠ 0.
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 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.
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