Reciprocal rule

Topics in Calculus
Differentiation
	Product rule; Quotient rule; Chain rule; Change of variables; Implicit differentiation; Taylor's theorem; Related rates; Identities
Integration
	Lists of integrals; Improper integrals; Integration by:; parts, disks, cylindrical; shells, substitution,; trigonometric substitution,; partial fractions, changing order
Vector calculus
	Gradient; Divergence; Curl; Laplacian; Gradient theorem; Green's theorem; Stokes' theorem; Divergence theorem
Multivariable calculus
	Matrix calculus; Partial derivative; Multiple integral; Line integral; Surface integral; Volume integral; Jacobian

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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

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

where $g(x) \neq 0.$

[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.