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Mathematics-Online course: Prepcourse Mathematics - Analysis - Differential Calculus

Quotient Rule

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The derivative of a quotient of two differentiable functions is given by

$\displaystyle \left( \frac{f}{g} \right)'=\frac{f'g-fg'}{g^2} $

in all $ x$ with $ g(x)\neq 0$. In particular, it is

$\displaystyle \left( \frac{1}{g} \right)'= -\frac{g'}{g^2}\,. $

(Authors: Höllig/Kopf/Abele)
The derivative of the rational function

$\displaystyle \frac{f(x)}{g(x)} = \frac{1-2x}{4+3x^2} $

is

$\displaystyle \frac{(1-2x)'(4+3x^2) - (1-2x)(4+3x^2)'}{(4+3x^2)^2} = \frac{-2(4+3x^2) - (1-2x)(6x)}{(4+3x^2)^2} = \frac{6x^2-6x-8}{(4+3x^2)^2} $

Alternatively, the product rule yields

$\displaystyle \left( (1-2x) \frac{1}{4+3x^2} \right)' = (-2) \frac{1}{4+3x^2} + (1-2x) \frac{-6x}{(4+3x^2)^2}= \frac{6x^2-6x-8}{(4+3x^2)^2}\,. $

(Authors: Höllig/Kreitz/Abele )
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  automatically generated 9/18/2007