Product and Quotient Rules
The product and quotient rules are:
\begin{eqnarray*} {\displaystyle \frac{d}{dx} \Bigl(f(x)g(x)\Bigr)} &=&
f(x)g'(x) + f'(x)g(x) \cr\cr\cr\cr\cr\cr\cr
\displaystyle{\frac{d}{dx}\left( \frac{f(x)}{g(x)}\right)} &=&
\displaystyle{\frac{g(x) \,f'(x)  f(x) g'(x)}{g(x)^2}}
\end{eqnarray*}

These are sometimes expressed in terms of $u$ and $v$, as
\begin{eqnarray*}
\displaystyle{\frac{d}{dx}}\bigl(uv\bigr)& =& \displaystyle{u \frac{dv}{dx}
+ \frac{du}{dx}v} \cr\cr\cr\cr\cr\cr
\displaystyle{\frac{d}{dx}\left ( \frac{u}{v}\right)} &=&
\displaystyle{\frac{v \frac{du}{dx}  u \frac{dv}{dx}}{v^2}}
\end{eqnarray*}

Notice that the $\displaystyle v \frac{du}{dx}$ term is positive and the $\displaystyle u \frac{dv}{dx}$ term
is negative. The way to remember that is that, if $u$ and $v$ are both
positive, then increasing the numerator $u$ will increase the ratio
$u/v$, while increasing the denominator $v$ will decrease the ration
$u/v$.
Example 1: To take the derivative of the product $x^2\sin(x)$,
let $u=x^2$ and $v=\sin(x)$. Then $u' = 2x$ and $v'=\cos(x)$,
hence $$\frac{d}{dx}\bigl(x^2\sin(x)\bigr)=\frac{d}{dx}\bigl(uv\bigr)=x^2\cos(x) + 2x\sin(x).$$

Example 2: To take the derivative of $\displaystyle \tan(x) = \frac{\sin(x)}{\cos(x)}$,
let $u=\sin(x)$ and $v=\cos(x)$, so $u'=\cos(x)$ and $v'=\sin(x)$.
Then $$\frac{d}{dx}\left ( \frac{u}{v}\right) = \frac{v u'  u v'}{v^2} =
\frac{ \cos(x)\cos(x)  \bigl(\sin(x)\bigr)\bigl(\sin(x)\bigr)}{\cos^2(x)},$$
which simplifies to $\displaystyle\frac{1}{\cos^2(x)} = \sec^2(x)$, since $\sin^2(x)+\cos^2(x)=1$. 
