Calculus I · 2A · review

Chain Rule Review

Review chain rule review with mixed practice and links back to the exact lessons behind each skill.

Course progress35 of 49 core pages35 of 49

Section overview

The chain rule and compositions

What this section is building

Review chain rule review with mixed practice and links back to the exact lessons behind each skill.

Notice

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

Decide

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

Avoid

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

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.

Source & rights

Original instruction with traceable references.

BetterGrades-original composition declared by source handoff; owner provenance review required before public release

Reference textbooks remain rights-separated and are not published as application assets. Any direct adaptation requires separate identification and attribution.

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.

Source & rights

Original instruction with traceable references.

BetterGrades-original composition declared by source handoff; owner provenance review required before public release

Reference textbooks remain rights-separated and are not published as application assets. Any direct adaptation requires separate identification and attribution.

Chain Rule Review

Related resources

Original instruction with traceable references.

Chain Rule Review

Related resources

Original instruction with traceable references.