Calculus I · 2A · review

Core Differentiation Rules Review

Review core differentiation rules review with mixed practice and links back to the exact lessons behind each skill.

Course progress19 of 49 core pages19 of 49

Section overview

Core differentiation rules

What this section is building

Review core differentiation rules review with mixed practice and links back to the exact lessons behind each skill.

Notice

Sums contribute independently, products have two changing contributions, and quotients must account for a changing denominator.

Decide

Name the outermost algebraic structure, then apply the smallest rule set that preserves it.

Avoid

Applying a familiar rule to the wrong outer structure or simplifying after a differentiation error.

Core Rules Review

Summary

(c)'=0, \qquad (x^n)'=nx^{n-1},

(af+bg)'=af'+bg',

(fg)'=f'g+fg', \qquad \left(\frac fg\right)'=\frac{f'g-fg'}{g^2}.

Rewrite radicals and reciprocals as powers. Simplify before using a more complicated rule.

Exercise

Differentiate $7x^8-3x^{1/2}+5x^{-4}$ .

Exercise

Differentiate $(x^3+2x)(x^2-1)$ .

Exercise

Differentiate $(x^2+4)/(x-1)$ .

Exercise

Find the tangent line to $y=x^4-2x^2$ at $x=1$ .

Exercise

Find all inputs where the tangent to $y=x^3-3x$ is horizontal.

Exercise

Differentiate $x^2/(x^3+1)$ and state its domain.

Exercise

A square has side $s(t)=t^2+1$ . Find the rate of change of its area at $t=2$ .

Exercise

Identify and repair the error: $\frac{d}{dx}[(x^2+1)(x^3-1)]=2x(3x^2)$ .

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.

Core Rules Review

Summary

(c)'=0, \qquad (x^n)'=nx^{n-1},

(af+bg)'=af'+bg',

(fg)'=f'g+fg', \qquad \left(\frac fg\right)'=\frac{f'g-fg'}{g^2}.

Rewrite radicals and reciprocals as powers. Simplify before using a more complicated rule.

Exercise

Differentiate $7x^8-3x^{1/2}+5x^{-4}$ .

Exercise

Differentiate $(x^3+2x)(x^2-1)$ .

Exercise

Differentiate $(x^2+4)/(x-1)$ .

Exercise

Find the tangent line to $y=x^4-2x^2$ at $x=1$ .

Exercise

Find all inputs where the tangent to $y=x^3-3x$ is horizontal.

Exercise

Differentiate $x^2/(x^3+1)$ and state its domain.

Exercise

A square has side $s(t)=t^2+1$ . Find the rate of change of its area at $t=2$ .

Exercise

Identify and repair the error: $\frac{d}{dx}[(x^2+1)(x^3-1)]=2x(3x^2)$ .

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.

Core Rules Review

Related resources

Original instruction with traceable references.

Core Rules Review

Related resources

Original instruction with traceable references.