Unit 2: Differentiation, Definition & Basic Rules | AP® Calculus AB Review

1. The definition of the derivative

The derivative is a limit. Two equivalent forms, both fair game on the exam:

$f'(x) = \lim_{h \to 0} \dfrac{f(x+h) - f(x)}{h} \qquad\qquad f'(a) = \lim_{x \to a} \dfrac{f(x) - f(a)}{x - a}$

The exam loves showing you one of these in disguise and asking for the limit. Before grinding algebra on any limit shaped like a difference quotient, ask: is this just $f'(a)$ for some familiar $f$?

Worked example

Evaluate $\lim_{h \to 0} \dfrac{(2+h)^5 - 2^5}{h}$.

Pattern-match: this is $f'(2)$ for $f(x) = x^5$.
$f'(x) = 5x^4$.
$f'(2) = 5 \cdot 16 = 80$. No expansion needed.

Common mistake: expanding $(2+h)^5$ by brute force. It works, but it burns five minutes on a question designed to take twenty seconds.

2. Slope, tangent lines, and rates of change

$f'(a)$ is three things at once: the slope of the tangent line at $x = a$, the instantaneous rate of change of $f$ at $a$, and the best linear approximation's slope. For motion, position's derivative is velocity and velocity's derivative is acceleration.

Tangent line recipe: find the point $(a, f(a))$, find the slope $f'(a)$, write $y - f(a) = f'(a)(x - a)$. The normal line uses the negative reciprocal slope.

Worked example

Find the tangent line to $y = x^3 - 2x$ at $(1, -1)$.

$y' = 3x^2 - 2$, so the slope at $x = 1$ is $1$.
Point-slope: $y + 1 = 1(x - 1)$.
$y = x - 2$.

Common mistake: plugging the x-value into $f$ when the question wants $f'$, or vice versa. Slope questions want the derivative; "the value of the function" wants $f$.

3. The power rule and linearity

$\dfrac{d}{dx} x^n = n x^{n-1}$ for every real $n$. Combined with "constants ride along" and "differentiate sums term by term," this handles all polynomials, plus roots and reciprocals once you rewrite them:

$\sqrt{x} = x^{1/2} \to \dfrac{1}{2\sqrt{x}} \qquad\qquad \dfrac{1}{x} = x^{-1} \to -\dfrac{1}{x^2}$

Worked example

Differentiate $f(x) = \dfrac{x^3 + 1}{x}$.

Simplify first: $f(x) = x^2 + x^{-1}$.
Power rule per term: $f'(x) = 2x - x^{-2}$.
Rewrite: $2x - \dfrac{1}{x^2}$. Simplifying beat the quotient rule.

Common mistake: reaching for the quotient rule when algebra would split the fraction. More rules means more chances for sign errors.

4. Product and quotient rules

$(uv)' = u'v + uv' \qquad\qquad \left(\dfrac{u}{v}\right)' = \dfrac{u'v - uv'}{v^2}$

The product rule is symmetric; the quotient rule is not. The numerator's subtraction order matters: top-prime-times-bottom comes first. These rules also work on pure numbers from a table, no formulas needed.

Worked example (table values)

Given $f(2) = 3$, $f'(2) = -1$, $g(2) = 5$, $g'(2) = 2$, find $(fg)'(2)$.

$(fg)' = f'g + fg'$.
$(fg)'(2) = (-1)(5) + (3)(2)$.
$= -5 + 6 = 1$.

Common mistake: "the derivative of a product is the product of the derivatives." It never is. Same for quotients.

5. Trig derivatives

All six, and the pattern that organizes them: every function starting with "c" (cos, cot, csc) carries a negative derivative.

$\sin \to \cos \qquad \cos \to -\sin \qquad \tan \to \sec^2$

$\sec \to \sec\tan \qquad \cot \to -\csc^2 \qquad \csc \to -\csc\cot$

Worked example

Find $f'(0)$ for $f(x) = \sec x$.

$f'(x) = \sec x \tan x$.
$\sec 0 = 1$, $\tan 0 = 0$.
$f'(0) = 0$.

Common mistake: dropping the negative on the three co-functions. The "starts with c" mnemonic exists because this error is that common.

6. Exponentials and logarithms

$\dfrac{d}{dx} e^x = e^x \qquad\qquad \dfrac{d}{dx} \ln x = \dfrac{1}{x}$

$e^x$ being its own derivative is what makes $e$ special. These two combine constantly with the product rule: $x e^x$, $x^2 \ln x$, and friends are exam staples.

Worked example

Differentiate $y = x^2 \ln x$.

Product rule: $y' = 2x \ln x + x^2 \cdot \dfrac{1}{x}$.
Simplify the second term: $x^2 / x = x$.
$y' = 2x\ln x + x$.

Common mistake: treating $e^x$ like a power function and writing $x e^{x-1}$. The power rule needs a constant exponent.

7. When derivatives fail

Differentiable means the graph is locally smooth: zoom in far enough and it looks like a line. Three classic failures, all continuous:

Corner ($|x|$ at $0$: one-sided slopes disagree), cusp (slopes blow up in opposite directions), vertical tangent ($x^{1/3}$ at $0$: slope goes infinite).

The one-way street to memorize: differentiable implies continuous, never the reverse. For piecewise functions to be differentiable at a seam, you need both continuity and matching slopes.

Worked example

Find $m$ and $b$ making $f(x) = \begin{cases} x^2 & x \leq 1 \\ mx + b & x > 1 \end{cases}$ differentiable everywhere.

Match slopes at $x = 1$: $2x = 2$, so $m = 2$.
Match values: $1^2 = 2(1) + b$, so $b = -1$.
Both conditions are required. Slopes alone are not enough.

Common mistake: matching slopes but forgetting continuity (or the reverse). Differentiability at a seam is a two-part check, and the exam tests both parts.

Unit 2 practice set: 45 questions

Every question exam-style, every solution step-by-step, with a topic report at the end.