Quotient rule

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

$f'(x) = \frac{h(x)g'(x) - g(x)h'(x)}{[h(x)]^2}.$

More precisely, if all x in some open set containing the number a satisfy $h(a)\not=0$, and $g'(a)$ and $h'(a)$ both exist, then $f'(a)$ exists as well and

$f'(a)=\frac{g'(a)h(a) - g(a)h'(a)}{[h(a)]^2}.$

And this can be extended to calculate the second derivative as well (one can prove this by taking the derivative of $f(x)=g(x)/h(x)$ twice). The result of this is:

$f''(x)=\frac{g''(x)[h(x)]^2-2g'(x)h(x)h'(x)+g(x)[2[h'(x)]^2-h(x)h''(x)]}{[h(x)]^3}.$

which can also be written in Lagrange's notation as

$y'' = \frac{v(vu''-uv'')-2v'(vu'-uv')}{{v}^3}$

The quotient rule formula can be derived from the product rule and chain rule.

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)/x² (when x ≠ 0) is:

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

Proof

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

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

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

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

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

Alternative proof (logarithmic differentiation)

Let $f = \frac{u}{v}$

$\ln f = \ln u - \ln v$

Differentiate both sides,

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

$\frac{f'}{u/v} = \frac{u'v-uv'}{uv}$

$f'= \frac{u'v-uv'}{uv} \cdot \frac{u}{v}$

$f' = \frac{u'v-uv'}{v^2}$

References

1. Stewart, James (2008). Calculus: Early Transcendentals (6th ed.). Brooks/Cole. ISBN 0-495-01166-5.
2. Larson, Ron; Edwards, Bruce H. (2009). Calculus (9th ed.). Brooks/Cole. ISBN 0-547-16702-4.
3. Thomas, George B.; Weir, Maurice D.; Hass, Joel (2010). Thomas' Calculus: Early Transcendentals (12th ed.). Addison-Wesley. ISBN 0-321-58876-2.