## derivation of the quotient rule

The quotient rule is used to take the derivative of a function with divided expressions:

$$\left(\frac{u}{v}\right)' = \frac{vu' - uv'}{v^2}$$

It is possible to prove this rule by utilizing the definition of the derivative; however, this is not nearly as elegant as the following simple proofs which use other derivative properties instead.

### product rule

\begin{align} y & = \frac{u}{v} \ & = uv^{-1} \ y' & = v^{-1}u' + u(-v^{-2}v') \ & = \frac{u'}{v} - \frac{uv'}{v^2} \ & = \frac{v}{v}\cdot\frac{u'}{v} - \frac{uv'}{v^2} \ & = \frac{vu'}{v^2} - \frac{uv'}{v^2} \ & = \frac{vu' - uv'}{v^2} \ \end{align}

### logarithm

\begin{align} y & = \frac{u}{v} \ \mathrm{ln} y & = \mathrm{ln} \frac{u}{v} \ & = \mathrm{ln} u - \mathrm{ln} v \ \frac{y'}{y} & = \frac{u'}{u} - \frac{v'}{v} \ & = \frac{v}{v}\frac{u'}{u} - \frac{u}{u}\frac{v'}{v} \ & = \frac{vu' - uv'}{uv} \ y' & = y\frac{vu' - uv'}{uv} \ & = \frac{u}{v}\frac{vu' - uv'}{uv} \ & = \frac{vu' - uv'}{v^2} \ \end{align}

## logarithmic proofs

As well as being used in the proof of the quotient rule, logarithms can also be used to prove a couple of other derivative rules.

### power rule

\begin{align} y & = x^n \ \mathrm{ln} y & = \mathrm{ln} x^n \ & = n \mathrm{ln} x \ \frac{y'}{y} & = n \frac{1}{x} \ y' & = yn \frac{1}{x} \ & = n x^n x^{-1} \ & = nx^{n-1}\ \end{align}

### product rule

\begin{align} y & = uv \ \mathrm{ln} y & = \mathrm{ln} uv \ & = \mathrm{ln} u + \mathrm{ln} v \ \frac{y'}{y} & = \frac{u'}{u} + \frac{v'}{v} \ y' & = y \frac{u'}{u} + \frac{v'}{v} \ & = uv \left(\frac{u'}{u} + \frac{v'}{v}\right) \ & = vu' + uv' \ \end{align}