Reciprocal rule
From Wikipedia, the free encyclopedia
- 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 <math>1/g(x)</math> is given by
- <math>\frac{d}{dx}\left(\frac{1}{g(x)}\right) = \frac{- g'(x)}{(g(x))^2}</math>
where <math>g(x) \neq 0.</math>
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[edit] Proof
[edit] From the quotient rule
The reciprocal rule is derived from the quotient rule, with the numerator <math>f(x) = 1</math>. Then,
<math>\frac{d}{dx}\left(\frac{1}{g(x)}\right) = \frac{d}{dx}\left(\frac{f(x)}{g(x)}\right)</math> <math>= \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}</math> <math>= \frac{0\cdot g(x) - 1\cdot g'(x)}{(g(x))^2}</math> <math>= \frac{- g'(x)}{(g(x))^2}.</math>
[edit] From the chain rule
The reciprocal rule can also be derived from the chain rule. Let <math>f(x)=x^{-1}.</math> Then,
- <math>f(g(x))=(g(x))^{-1}=\frac{1}{g(x)}</math>.
By the chain rule,
<math>\frac{d}{dx}\left(\frac{1}{g(x)}\right) = \frac{d}{dx} f(g(x))</math> <math>= f'(g(x))\cdot g'(x)</math> <math>= -(g(x))^{-2} \cdot g'(x)</math> <math>= \frac{- g'(x)}{(g(x))^2}.</math>
[edit] Examples
The derivative of <math>1/(x^2 + 2x)</math> is:
- <math>\frac{d}{dx}\left(\frac{1}{x^2 + 2x}\right) = \frac{-2x - 2}{(x^2 + 2x)^2}.</math>
The derivative of <math>1/\cos(x)</math> (when <math>\cos x\not=0</math>) is:
- <math>\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).</math>
For more general examples, see the derivative article.
[edit] See also
- Product rule
- Quotient rule
- Chain rule
- Difference quotientit:Regola della funzione reciproca
zh:倒数定则
