Combining Rules
Many functions can't be cracked open with a single rule. Instead,
we can use a rule to break the problem down into (possibly
several) simpler problems. Then we can use another rule on
each of those. And repeat as long as needed to get the answer.
Example: Find $F'(x)$ where $$
F(x)=\frac{\sin\left(x^2\right)}{1 + e^{2x}}. $$
Solution: This is a quotient, so $$ F'(x) =
\frac{\left(1+e^{2x}\right)\frac{d}{dx}\Big((\sin\left(x^2\right)\Big)

\sin\left(x^2\right)\frac{d}{dx}\Bigl(1+e^{2x}\Bigr)}{\left(1+e^{2x}\right)^2},$$
by the quotient rule. But that still leaves us with the question
of how to compute the derivatives of $\sin\left(x^2\right)$ and
$\left(1+e^{2x}\right)$. Each of those can be computed using the
chain rule.
It can be dangerous to combine steps.
Until you've really got the hang of it, write out your
calculations with at most one rule per line. For instance, you
might write:
\begin{eqnarray*}F'(x) = &
\frac{(1+e^{2x})\frac{d}{dx}\Big(\sin\left(x^2\right)\Big) 
\sin\left(x^2\right)\frac{d}{dx}\Bigl(1+e^{2x}\Bigr)}{\left(1+e^{2x}\right)^2}
& \qquad \hbox{(by the quotient rule)} \cr= &
\frac{\left(1+e^{2x}\right)\left(\cos\left(x^2\right)\right)(2x) 
\sin\left(x^2\right)\frac{d}{dx}\Bigl(1+e^{2x}\Bigr)}{\left(1+e^{2x}\right)^2}
& \qquad \hbox{(by the chain rule applied to
}\sin\left(x^2\right)\hbox{)} \cr= &
\frac{\left(1+e^{2x}\right)\left(\cos\left(x^2\right)\right)(2x) 
\sin\left(x^2\right)e^{2x}\frac{d}{dx}\Bigl(2x\Bigr)}{\left(1+e^{2x}\right)^2}
& \qquad \hbox{(by the chain rule applied to }e^{2x}\hbox{)}
\cr= & \frac{2x\left(1+e^{2x}\right)\cos(x^2) 
2\sin\left(x^2\right)e^{2x}}{\left(1+e^{2x}\right)^2} & \qquad
\hbox{(by algebra)} \cr\end{eqnarray*}
Eventually you'll be able to
do several operations in one steps, but please practice first.
There isn't always an obvious order in which to apply the rules.
When in doubt, think about the structure of the function.
This is the big picture of the
function. Would you describe it as a product of two simpler
functions? If so, apply the product rule first. Would you
describe it as a quotient? If so, apply the quotient rule
first. Is it instead a power, or a log, or an exponential,
or a trig function of some complicated expression (which may
itself involve products, quotients, or further nesting)? If
so, apply the chain rule first.
