The Chain Rule

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The Chain Rule

Two Forms of the Chain Rule
Version 1
Version 2
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A hybrid chain rule

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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.