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29 yr old Orthopaedic Surgeon Grippo from Saint-Paul, spends time with interests including model railways, top property developers in singapore developers in singapore and dolls. Finished a cruise ship experience that included passing by Runic Stones and Church.

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

ddx(1g(x))=g(x)(g(x))2

where g(x)0.

Proof

From the quotient rule

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

ddx(1g(x))=ddx(f(x)g(x)) =f(x)g(x)f(x)g(x)(g(x))2
=0g(x)1g(x)(g(x))2
=g(x)(g(x))2.

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 1g(x) as being the function 1x composed with the function g(x). The result then follows by application of the chain rule.

Examples

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

ddx(1x3+4x)=3x24(x3+4x)2.

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

ddx(1cos(x))=sin(x)cos2(x)=1cos(x)sin(x)cos(x)=sec(x)tan(x).

For more general examples, see the derivative article.

See also

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