A hybrid Chain Rule
In addition to Versions 1 and 2 of the chain rule, there is also a
useful hybrid form:
$$\frac{d}{dx}f(u) = f'(u) \frac{du}{dx}$$

Applications
This form allows us to expand the scope of many of our derivative formulas:

The derivative of $x^n$ (with respect to $x$) is $n x^{n1}$, so the
derivative of $u^n$ is $n u^{n1} \frac{du}{dx}$.

The derivative of $\sin(x)$ is $\cos(x)$, so the derivative of
$\sin(u)$ is $\cos(u) \frac{du}{dx}$.

The derivative of $\cos(x)$ is $\sin(x)$,
so the derivative of $\cos(u)$ is $\sin(u) \frac{du}{dx}$.

The derivative of $e^x$ is $e^x$,
so the derivative of $e^u$ is $e^u \frac{du}{dx}$.
These formulas come up a lot, and are worth memorizing.
We can also use the chain rule to go back and find
derivatives of some functions we couldn't derive before.
Example: What is the derivative of $a^x$, where $a$ is
a positive number?
Solution:
Since $a= e^{\ln(a)}$, we can
rewrite $a^x$ as $(e^{\ln(a)})^x = e^{x \ln(a)}$. Taking $u=x \ln(a)$,
we have
\begin{eqnarray*}
\displaystyle{\frac{d a^x}{dx}} &=&
\displaystyle{\frac{d}{dx}(e^{x \ln(a)})} \cr
&=& e^{x \ln(a)} \displaystyle{\frac{d}{dx}}\Big(x\ln(a)\Big)\cr
& = & e^{x \ln(a)} \ln(a) \cr
& = & a^x \ln(a).
\end{eqnarray*}
(Notice that $\ln(a)$ is a constant.)

