Calculus I · 2A · reference

Unit 2A Formula and Strategy Reference

Use this derivative foundations and techniques reference for formulas, notation, domains, units, and problem-solving strategy.

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Section overview

Review, practice, exams, and reference

What this section is building

Use this derivative foundations and techniques reference for formulas, notation, domains, units, and problem-solving strategy.

Notice

A dependable solution separates interpretation, rule selection, calculation, and verification.

Decide

Classify first, calculate second, and use answer reveals to diagnose the first incorrect decision.

Avoid

Reading the solution before making a complete attempt or treating every miss as mere algebra.

Formula and Strategy Reference

f'(a)=\lim_{h\to0}\frac{f(a+h)-f(a)}h, \qquad f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}h.

{2}

\begin{aligned} \frac{d}{dx}c&=0,& \frac{d}{dx}x^n&=nx^{n-1},\\ (cf)'&=cf',& (f\pm g)'&=f'\pm g',\\ (fg)'&=f'g+fg',& \left(\frac fg\right)'&=\frac{f'g-fg'}{g^2},\\ (f\circ g)'&=f'(g)g',& (\sin x)'&=\cos x,\\ (\cos x)'&=-\sin x,& (\tan x)'&=\sec^2x,\\ (e^x)'&=e^x,& (a^x)'&=a^x\ln a,\\ (\ln x)'&=1/x,& (\log_a x)'&=1/(x\ln a),\\ (\arcsin x)'&=1/\sqrt{1-x^2},& (\arctan x)'&=1/(1+x^2). \end{aligned}

For an inverse function,

(f^{-1})'(b)=\frac1{f'(a)}\quad\text{when }f(a)=b.

For implicit differentiation, differentiate both sides with respect to $x$ and include $y'$ for each differentiated $y$ -term. For variable exponents, take logarithms.

Bridge

Continue to Unit 2B

Unit 2A ends with the ability to calculate and explain derivatives. Unit 2B begins by using those derivatives to analyze motion, approximation, related rates, graph shape, optimization, and indeterminate limits.

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