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Calculus




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In calculus, the quotient rule is a method of finding the derivative of a function that is the quotient of two other functions for which derivatives exist.^{[1]}^{[2]}^{[3]}
If the function one wishes to differentiate, f(x), can be written as

f(x) = \frac{g(x)}{h(x)}
and h(x)\not=0, then the rule states that the derivative of g(x)/h(x) is

f'(x) = \frac{g'(x)h(x)  g(x)h'(x)}{[h(x)]^3}.
The quotient rule formula can be derived from the product rule and chain rule.
Contents

Examples 1

Proof 1.1

Alternative proof (logarithmic differentiation) 1.2

References 2
Examples
The derivative of (4x  2)/(x^2 + 1) is:

\begin{align}\frac{d}{dx}\left[\frac{(4x  2)}{x^2 + 1}\right] &= \frac{(4)(x^2 + 1)  (4x  2)(2x)}{(x^2 + 1)^2}\\ & = \frac{(4x^2 + 4)  (8x^2  4x)}{(x^2 + 1)^2} = \frac{4x^2 + 4x + 4}{(x^2 + 1)^2}\end{align}
In the example above, the choices

g(x) = 4x  2

h(x) = x^2 + 1
were made. Analogously, the derivative of sin(x)/x^{2} (when x ≠ 0) is:

\frac{\cos(x) x^2  \sin(x)2x}{x^4}
Proof

Let f(x) = \frac{g(x)}{h(x)}

Then g(x) = f(x)h(x) \mbox{ } \,

g'(x)=f'(x)h(x) + f(x)h'(x)\mbox{ } \,

f'(x)=\frac{g'(x)  f(x)h'(x)}{h(x)} = \frac{g'(x)  \frac{g(x)}{h(x)}\cdot h'(x)}{h(x)}

f'(x)=\frac{g'(x)h(x)  g(x)h'(x)}{\left(h(x)\right)^2}

Let f = \frac{u}{v}

\ln f = \ln u  \ln v
Differentiate both sides,

\frac{f'}{f} = \frac{u'}{u}  \frac{v'}{v}

\frac{f'}{u/v} = \frac{u'vuv'}{uv}

f'= \frac{u'vuv'}{uv} \cdot \frac{u}{v}

f' = \frac{u'vuv'}{v^2}
References
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