Calculus I · Unit 2A · reference

Unit 2A Formula and Strategy Reference

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.

After the explanation

Use the section idea

Reading lens

Mixed practice tests recognition: the page title no longer tells you which rule to use.

Mental model

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

Decision

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

Common trap

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

Check yourself

Can you explain why your method fits before comparing your final answer?

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