A cute (and sometimes useful) trick is \Def{logarithmic differentiation}:
take $\log$ of $f$, then differentiate, which gives you
$\frac{f'}{f}$.
This is useful for an assortment of purposes, for instance in complex
analysis.

Here's one:
if $f=f_1f_2\dots f_n$ (it's a big product),
then $\log f = \log f_1 + \dots + \log f_n$,
so
$$\frac{f'}{f}=\frac{f'_1}{f_1}+\dots \frac{f'_n}{f_n}$$
and so
$$f'=f\left(\frac{f'_1}{f_1}+\dots \frac{f'_n}{f_n}\right)$$
(though this is actually pretty obvious: it's just a cute way of writing the
product rule:
$$f'=f'_1f_2\dots f_n + f_1 f'_2 f_3\dots f_n + \dots + f_1\dots f_{n-1} f'_n$$

I think it complex analysis this lets you do cute things with poles or such.

