Calculus I · Unit 2A · review

Chain Rule Review

Summary

For $y=F(g(x))$ ,

y'=F'(g(x))g'(x).

Differentiate from the outside inward. One derivative factor is required for every changing layer. Product and quotient rules may surround chain-rule derivatives.

Exercise

Differentiate $(5x^2-1)^7$ .

Exercise

Differentiate $\sqrt{1+x^4}$ .

Exercise

Differentiate $\cos(3x^2)$ .

Exercise

Differentiate $e^{\sin x}$ .

Exercise

Differentiate $\ln|x^3-4x|$ .

Exercise

Differentiate $x^3\tan(2x)$ .

Exercise

Differentiate $\sin(x^2)/(x^2+1)$ .

Exercise

Differentiate $[1+e^{-x}]^{-1}$ , the standard logistic form.

After the explanation

Use the section idea

Reading lens

Read nested functions from the outside inward, but multiply local response factors through every layer.

Mental model

A small input change passes through a sequence of machines; the total response multiplies the response at each stage.

Decision

List the layers, differentiate one layer at a time, and stop only when every input-dependent layer contributes.

Common trap

Differentiating the outside and leaving the inside unchanged without its derivative factor.

Check yourself

Can you annotate every factor in your derivative with the layer that produced it?

Source & rights

Original instruction with traceable references.

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