Two Forms of the Chain Rule
The chain rule is one of the most powerful tools for computing
derivatives. It allows us to differentiate composite functions. There are two
forms of it:
 If $f$ and $g$ differentiable functions, then $$
\Big(f\big(g(x)\big)\Big)' = f'\big(g(x)\big) \cdot
g'(x).$$
 If $y=f(u)$ and $u=g(x)$, then $$\frac{dy}{dx} =
\frac{dy}{du} \frac{du}{dx}.$$

The two versions mean the exact same thing, but sometimes it's
easier to think in terms of one or the other. The first version is
best for computing derivatives of expressions like $(5+3x)^5$ or
$\ln(3+\cos(x))$.
One helpful way to use the first version of the chain rule is to
find the "inside part" of the composite function $f(g(x))$, which
is $g(x)$.
DO: What is the inside
part of $(5+3x)^5$?
Then find the "outside part", which is $f(\quad)$. The
outside part of $(5+3x)^5$ is $f(u)=(u)^5$. Then $f'(u)$ is
$5u^4$ and so the derivative of the composition is $5(\text{inside
part})^4\cdot\frac{d}{dx}(\text{inside part})=5(5x+3)^4\cdot(5)$.
DO: What is the inside
part of $\ln(3+\cos(x))$? What is the outside part?
You will get much practice with this.
The second version is best for understanding related rates or
logarithmic derivatives.
