Rahul Sharma (Editor)

Reciprocal rule

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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.

Contents

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

d d x ( 1 g ( x ) ) = − g ′ ( x ) ( g ( x ) ) 2

where g(x) ≠ 0.

From the quotient rule

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

d d x ( 1 g ( x ) ) = d d x ( f ( x ) g ( x ) ) = f ′ ( x ) g ( x ) − f ( x ) g ′ ( x ) ( g ( x ) ) 2 = 0 ⋅ g ( x ) − 1 ⋅ g ′ ( x ) ( g ( x ) ) 2 = − g ′ ( x ) ( g ( x ) ) 2 .

From the chain rule and power rule

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.

d d x ( 1 g ( x ) ) = d d x ( g ( x ) ) − 1 = − 1 ⋅ ( g ( x ) ) − 2 ⋅ g ′ ( x ) = − g ′ ( x ) ( g ( x ) ) 2

Examples

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

d d x ( 1 x 3 + 4 x ) = − 3 x 2 − 4 ( x 3 + 4 x ) 2 .

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

d d x ( 1 cos ⁡ ( x ) ) = sin ⁡ ( x ) cos 2 ⁡ ( x ) = 1 cos ⁡ ( x ) sin ⁡ ( x ) cos ⁡ ( x ) = sec ⁡ ( x ) tan ⁡ ( x ) .

For more general examples, see the derivative article.

References

Reciprocal rule Wikipedia
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