Products and quotients: when two functions are combined

Name the two pieces $u$ and $v$ (each is a function of $x$).

Product rule, for $y = uv$:

$\frac{dy}{dx} = u\frac{dv}{dx} + v\frac{du}{dx}$

In words: first times derivative of second, plus second times derivative of first.

Shorthand: $u'$ means $\dfrac{du}{dx}$, so the product rule is $(uv)' = uv' + vu'$.

Quotient rule, for $y = \dfrac{u}{v}$ (with $u$ on top and $v$ on the bottom):

$\frac{dy}{dx} = \frac{v\dfrac{du}{dx} - u\dfrac{dv}{dx}}{v^2}$

In words: bottom times derivative of top, minus top times derivative of bottom, all over the bottom squared.

Order matters in the quotient rule because of the minus sign. In the product rule it doesn't: swapping $u$ and $v$ gives the same answer.