Calculus I · Unit 2A · review

Core Differentiation Rules Review

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

After the explanation

Use the section idea

Reading lens

Every rule is a compressed limit calculation; choose the structure before doing algebra.

Mental model

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

Decision

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

Common trap

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

Check yourself

Can you justify the primary rule before writing the first derivative symbol?

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