The basic derivative rules can be understood geometrically:

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linearity is obvious; you can also see it by dilating a rod or
concatenating two rods

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the product rule is cute:
given a rectangle of sides $f(x),g(x)$, the area is $f(x)g(x)$,
and the area changes by $f'(x)g(x) + f(x)g'(x)$
(draw picture: there's a little $h^2$ in the corner that goes away)

(it's better to see by seeing the derivative of $xy$ in two variables:
$x\dy + y\dx$, but the picture's the same)

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the power rule is similar, in $n$-dimensions:
the $n$-cube has $n$ faces which are increasing, each by $x^{n-1}$,
hence $(x^n)'=nx^{n-1}$.

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the chain rule is best understood by first doing composition of
-linear- functions.
this I visualize by having three gears: if you turn the first one once,
the second turns by $M$; if you turn the second once, the third turns
by $N$; so if you turn the first once, the third turns by $MN$:
tada: composition is multiplication, for linear stuff.

I also like the $X \to Y \to Z$ picture: composition does what it
should (note that it really should be:
$(gf)'(x) = f'(x)\cdot g'(y)$, where $y=f(x)$, so computationally
$(gf)'(x) = g'(f(x))\cdot f'(x)$:
it's ``peeling the onion'' (don't you -dare- touch what's inside: it's
holy. Only when you "throw it out of heaven" can you differentiate it)

I prove the chaing rule by saying $f(x+h)=f(x) + f'(x)h + o(h)$
and composing.

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the quotient rule is best understood as coming from a combination:
$f/g = f\cdot i(g)$, where $i(g)=1/g$. Then the derivative of the
quotient is the product rule, together with the derivative of $1/x$,
together with the chain rule -- that's why it's so ugly.
Just memorize it.

mneumonics
``lo d hi - hi d lo (over lo squared)''
(bottom times derivative of top, minus other way, over bottom squared)

