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Unit 2 Cheat Sheet

Differentiation — every formula, every decision rule, and every trap from the five lessons, on one printable page. Each block title links back to its lesson.
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The Derivative Defined 2.1 – 2.3

The derivative is the instantaneous rate of change — the slope of the tangent line, defined as a limit of secant slopes.

limit definition
alternate form (at a point)
estimate from a table
Reading a limit as a derivative
\(\lim_{h\to0}\frac{f(a+h)-f(a)}{h}\)\(f'(a)\) — the added value is \(a\)
\(\lim_{x\to a}\frac{f(x)-f(a)}{x-a}\)\(f'(a)\) — alternate form
\(\tfrac00\) at a pointa derivative in disguise — factor & cancel
trap
  • Slope, not value. \(f'(a)\) is a rate; \(f(a)\) is a height. The standalone constant in the limit is \(f(a)\), the added term is \(a\).
  • Divide by the run. A table estimate is \(\frac{\Delta f}{\Delta x}\) — never just \(\Delta f\).

Differentiability & Continuity 2.4

Differentiable means “locally straight.” It forces continuity — but continuity does not force differentiability.

the one-way implication
piecewise: differentiable at a seam
The four ways a derivative dies
corner (e.g. \(|x|\))one-sided slopes differ — finite mismatch
cusp (e.g. \(x^{2/3}\))slopes run to \(\pm\infty\)
vertical tangent (e.g. \(\sqrt[3]{x}\))slope infinite
discontinuity (jump/hole)not even continuous — auto-fail
trap
  • Piecewise needs TWO tests. Continuous (pieces meet) and smooth (one-sided slopes match). Continuity alone isn't enough.
  • Corner = continuous. A corner breaks differentiability, not continuity. Only a jump/hole breaks continuity.

Power Rule, Tangents & Normals 2.5 – 2.6

One rule for every power, plus linearity for term-by-term differentiation — then the payoffs: tangent lines, horizontal tangents, normals.

power rule (any real \(n\))
constant, multiple, sum
tangent line at \(x=a\)
normal line (perpendicular)
Tangent-line tasks
“horizontal tangent”solve \(f'(x)=0\)
“tangent at \(x=a\)”point \(f(a)\) + slope \(f'(a)\), point-slope
“normal line”slope \(=-1/f'(a)\)
trap
  • Rewrite first. Roots and reciprocals become exponents: \(\sqrt{x}=x^{1/2}\), \(\frac1{x^n}=x^{-n}\). Watch the sign on negative exponents.
  • Not on \(2^x\). The power rule is for variable base, constant exponent — never when the variable is the exponent.
  • Horizontal tangent: \(f'=0\), not \(f=0\).

Key-Function Derivatives 2.7, 2.10

The eight you memorize — but they're full of pattern: the sin/cos cycle, \(e^x\) unchanged, and the co-functions' minus signs.

trig (the core two)
exponential & log
the other four trig
The sin/cos cycle (period 4)
\(\sin\to\cos\to-\sin\to-\cos\to\sin\)reduce the order mod 4
“co-” functions (cos, cot, csc)derivatives carry a minus sign
tan/cotgive squares (\(\sec^2,-\csc^2\))
sec/cscgive products (\(\sec\tan,-\csc\cot\))
trap
  • \(\cos\) gets the minus. \(\frac{d}{dx}\cos x=-\sin x\). The single most common sign slip.
  • \(\frac{d}{dx}\ln x=\frac1x\), not \(\ln x\) or \(x\). And \(e^x\) stays \(e^x\) — never \(xe^{x-1}\).

Product & Quotient Rules 2.8 – 2.9

For products and quotients, you can't differentiate the pieces separately — both factors change at once.

product rule
quotient rule
Which approach?
polynomial \(\times\) polynomialexpand first, then power rule
non-polynomial factor (trig/exp/log)product rule
quotient that simplifiescancel first (e.g. \(\frac{x^2}{x}=x\))
quotient that doesn'tquotient rule, \(v\ne0\)
trap
  • \((uv)'\ne u'v'\). The product rule is \(u'v+uv'\) — the number-one error in the unit.
  • Quotient order matters. “Low d-high minus high d-low, over low squared”: \(\frac{u'v-uv'}{v^2}\). Reversing flips the sign.
  • Square the denominator. It's \(v^2\), not \(v\).

The Eight Derivatives memorize

Your core toolkit. Every harder derivative is built from these by the power, product, quotient, and (Unit 3) chain rules.

power
sine / cosine
exponential / log
tangent / cotangent
secant / cosecant
∴

Every formula on this page is the same idea wearing different clothes: the derivative is the instantaneous rate of change — the slope of the tangent — defined as the limit of secant slopes. Unit 2 is the trade you make once and keep forever: you stop computing that limit by hand for every function and start reaching for rules instead. Power, sine/cosine, \(e^x\) and \(\ln x\), product, quotient — each one is a shortcut that the limit definition already proves. Master differentiability (when a derivative even exists) and these eight derivatives, and the rest of calculus is just combining them.

Test yourself on Unit 2 →
The Derivative Defined Differentiability Power & Basic Rules Key-Function Derivatives Product & Quotient