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