Chain Rule For Finding Derivatives

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0:20The chain rule is $[f(g(x))]'=f'(g(x))g'(x)$. Differentiate the outer function at the unchanged inner expression, then multiply by the inner derivative.

1:30For $(5x+3)^4$, the outside derivative contributes $4(5x+3)^3$ and the inside derivative contributes 5. Their product is $20(5x+3)^3$.

3:00The same rule gives $[\cos(x^2)]'=-2x\sin(x^2)$ and $[\tan(x^3)]'=3x^2\sec^2(x^3)$ wherever the original functions are defined.

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