Reciprocal rule

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This is about a method in calculus. For other uses of "reciprocal", see reciprocal.
Topics in calculus

Fundamental theorem
Limits of functions
Continuity
Vector calculus
Matrix calculus
Mean value theorem

Differentiation

Product rule
Quotient rule
Chain rule
Implicit differentiation
Taylor's theorem
Related rates
List of differentiation identities

Integration

Lists of integrals
Improper integrals
Integration by:
parts, disks, cylindrical
shells
, substitution,
trigonometric substitution,
partial fractions, changing order

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

\frac{d}{dx}\left(\frac{1}{g(x)}\right) = \frac{- g'(x)}{(g(x))^2}

where g(x) \neq 0.

Contents

[edit] Proof

[edit] From the quotient rule

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

\frac{d}{dx}\left(\frac{1}{g(x)}\right) = \frac{d}{dx}\left(\frac{f(x)}{g(x)}\right) = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}
= \frac{0\cdot g(x) - 1\cdot g'(x)}{(g(x))^2}
= \frac{- g'(x)}{(g(x))^2}.

[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 \frac{1}{g(x)} as being the function \frac{1}{x} 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:

\frac{d}{dx}\left(\frac{1}{x^2 + 2x}\right) = \frac{-2x - 2}{(x^2 + 2x)^2}.

The derivative of 1 / cos(x) (when \cos x\not=0) is:

\frac{d}{dx} \left(\frac{1}{\cos(x) }\right) = \frac{\sin(x)}{\cos^2(x)} = \frac{1}{\cos(x)} \frac{\sin(x)}{\cos(x)} = \sec(x)\tan(x).

For more general examples, see the derivative article.

[edit] See also

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