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.