AP Calculus AB/BC Unit 2 ReviewFundamentals of Differentiation

AP Calculus Unit 2, Fundamentals of Differentiation, is where you formally define the derivative as a limit of a difference quotient and learn the core rules for computing it, including the Power Rule, Product Rule, Quotient Rule, and the derivatives of sin x, cos x, e^x, and ln x. The single biggest idea is that the derivative measures an instantaneous rate of change, which is the slope of the tangent line at a point. Unit 2 makes up 10-12% of the AP exam, and every differentiation unit after this one builds directly on it.

What this unit covers

From average to instantaneous rate of change

The unit opens by upgrading slope from "between two points" to "at one point."

• The average rate of change of f on [a, b] is the slope of the secant line, $\frac{f(b)-f(a)}{b-a}$. It tells you how fast f changed on average over an interval.
• The instantaneous rate of change at x = a comes from shrinking that interval to zero. Formally, $f'(a) = \lim_{h \to 0} \frac{f(a+h)-f(a)}{h}$, or equivalently $\lim_{x \to a} \frac{f(x)-f(a)}{x-a}$.
• Picture secant lines pivoting toward the tangent line as the second point slides toward the first. The derivative is the slope they're approaching.
• The derivative of f as a whole function is $f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$. Notations you'll see everywhere are $f'(x)$, $\frac{dy}{dx}$, and $y'$.
• Since $f'(a)$ is the tangent slope, the tangent line at $(a, f(a))$ is $y = f(a) + f'(a)(x-a)$. Tangent line equations show up constantly on the exam.

Estimating derivatives from tables, graphs, and technology

Not every function comes with a formula, and the exam knows it.

• From a table, estimate $f'(a)$ with a difference quotient using the closest available x-values around a.
• From a graph, estimate the derivative by eyeballing the slope of the tangent line at the point.
• On calculator-allowed sections, your calculator can compute a numerical derivative at a point directly.

When derivatives exist (and when they don't)

This is the conceptual heart of the unit, and a favorite multiple choice target.

• Differentiability implies continuity. If $f'(a)$ exists, f is continuous at a. If a point isn't in the domain of f, it can't be in the domain of $f'$.
• The converse fails. A function can be continuous at a point and still not differentiable there.
• The classic failures are a corner, where the left and right limits of the difference quotient disagree, as with $f(x) = |x|$ at x = 0, and a vertical tangent, where the slope is undefined, as with $f(x) = \sqrt[3]{x}$ at x = 0.
• Any discontinuity (jump, hole, asymptote) automatically kills differentiability there.

The basic derivative rules

Once the definition is in place, these rules let you skip the limit work for whole families of functions.

• Power Rule: $\frac{d}{dx} x^r = r x^{r-1}$, which works for negative and fractional exponents too. Rewrite $\sqrt{x}$ as $x^{1/2}$ and $\frac{1}{x^2}$ as $x^{-2}$ first.
• Constant, Sum, Difference, and Constant Multiple Rules let you differentiate any polynomial term by term.
• Four derivatives to memorize cold: $\frac{d}{dx}\sin x = \cos x$, $\frac{d}{dx}\cos x = -\sin x$, $\frac{d}{dx} e^x = e^x$, $\frac{d}{dx}\ln x = \frac{1}{x}$.
• Bonus skill: recognizing the definition of the derivative inside a limit. A limit like $\lim_{h \to 0} \frac{\sin(\frac{\pi}{2}+h) - \sin(\frac{\pi}{2})}{h}$ is just $\cos(\frac{\pi}{2}) = 0$ in disguise.

Product Rule, Quotient Rule, and the other four trig functions

• Product Rule: $\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$. The derivative of a product is NOT the product of the derivatives.
• Quotient Rule: $\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{g(x)f'(x) - f(x)g'(x)}{[g(x)]^2}$. Order in the numerator matters.
• The remaining trig derivatives come from rewriting with identities. Since $\tan x = \frac{\sin x}{\cos x}$, the Quotient Rule gives $\frac{d}{dx}\tan x = \sec^2 x$, and similarly for cot, sec, and csc.

Unit 2, Fundamentals of Differentiation at a glance

Why Unit 2, Fundamentals of Differentiation matters in AP Calc

This unit answers the question that motivates half of calculus: how fast is something changing right now? Unit 1 built limits as a tool, and Unit 2 uses that tool to define the derivative. Everything in Units 3 through 5 is differentiation, and you can't do any of it without the rules introduced here.

• The derivative as a limit ties the course's big ideas together. Change (rates), limits, and the behavior of functions all meet in one definition.
• The differentiability-continuity relationship is one of the course's recurring "implication, not equivalence" themes, alongside ideas like continuity not guaranteeing differentiability.
• Fluency matters here in a way it doesn't elsewhere. If the Quotient Rule takes you ninety seconds, every later problem slows down. These rules need to be automatic by Unit 3.

How this unit connects across the course

• Limits and Continuity (Unit 1) supply the machinery. The derivative is literally a limit, and Topic 2.4 leans on Unit 1's definition of continuity to explain when derivatives fail to exist.
• Composite, Implicit, and Inverse Differentiation (Unit 3) extends this toolkit with the Chain Rule. The Product and Quotient Rules from Unit 2 get combined with the Chain Rule constantly there.
• Contextual and Analytical Applications (Units 4 and 5) put these derivatives to work in related rates, motion, optimization, and curve analysis. Every "find where f is increasing" problem starts by computing $f'$ with Unit 2 rules.
• Integration (Unit 6) reverses the process. Antiderivatives are built by running Unit 2's rules backward, so knowing $\frac{d}{dx} e^x = e^x$ instantly tells you $\int e^x\,dx = e^x + C$.

Key formulas and procedures

• Average rate of change: $\frac{f(b)-f(a)}{b-a}$. Slope of the secant line; use it for table-based "average rate" questions.
• Limit definition of the derivative: $f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$. Use it to derive derivatives from scratch or to recognize a derivative hiding inside a limit problem.
• Alternate form at a point: $f'(a) = \lim_{x \to a} \frac{f(x)-f(a)}{x-a}$. Same idea, different bookkeeping; the exam uses both.
• Tangent line equation: $y = f(a) + f'(a)(x - a)$. Point plus slope; this pattern returns in Unit 4 for linear approximation.
• Power Rule: $\frac{d}{dx} x^r = r x^{r-1}$. Works for all real exponents; rewrite radicals and reciprocals as powers first.
• Sum, Difference, Constant Multiple Rules: differentiate term by term and pull constants out front.
• Core derivatives: $\frac{d}{dx}\sin x = \cos x$, $\frac{d}{dx}\cos x = -\sin x$, $\frac{d}{dx}e^x = e^x$, $\frac{d}{dx}\ln x = \frac{1}{x}$.
• Product Rule: $(fg)' = f'g + fg'$. Use whenever two variable expressions are multiplied.
• Quotient Rule: $\left(\frac{f}{g}\right)' = \frac{gf' - fg'}{g^2}$. Use for ratios; check first whether algebraic simplification avoids it entirely.
• Trig derivatives from identities: $(\tan x)' = \sec^2 x$, $(\cot x)' = -\csc^2 x$, $(\sec x)' = \sec x \tan x$, $(\csc x)' = -\csc x \cot x$.
• Differentiability check at a point: confirm continuity first, then confirm the left and right limits of the difference quotient match (no corner, cusp, or vertical tangent).

Unit 2, Fundamentals of Differentiation on the AP exam

Unit 2 is 10-12% of the AP exam, but its skills appear in far more than 10-12% of the questions because later units depend on them. Expect to see this content in several forms:

• Multiple choice questions that hand you a limit and ask you to recognize it as a derivative. If you see $\lim_{h \to 0} \frac{f(a+h)-f(a)}{h}$ with a specific function plugged in, identify the function and the point, then evaluate the known derivative instead of computing the limit by hand.
• Computation questions that mix rules, like differentiating $\frac{x^2 \sin x}{e^x}$ or a polynomial with fractional exponents. Speed and accuracy with the Power, Product, and Quotient Rules are what's being tested.
• Conceptual questions on differentiability vs. continuity, often with a piecewise function. You may need to choose constants that make a piecewise function differentiable, which requires matching both function values and one-sided derivative values at the seam.
• Table and graph questions where you estimate a derivative from data using a difference quotient, a skill that also appears inside free response parts in later units (motion, rates from tables).
• Tangent line questions, both as standalone problems and as the first part of a free response, since the tangent line at a point only needs $f(a)$ and $f'(a)$.

Always use correct notation in free response work. Writing $f'(3)$ or $\frac{dy}{dx}\big|_{x=3}$ matters for earning points; a bare number without notation can cost you.

Essential questions

• How can we make "rate of change at a single instant" mathematically precise when slope normally requires two points?
• What does the derivative tell us about a function's graph, and what does the graph tell us about the derivative?
• Why does differentiability guarantee continuity, but continuity not guarantee differentiability?
• How do general rules let us differentiate complicated functions without returning to the limit definition every time?

Key terms to know

• Difference quotient: The expression $\frac{f(x+h)-f(x)}{h}$, whose limit as h approaches 0 defines the derivative.
• Average rate of change: The slope of the secant line through two points on a function, computed as $\frac{f(b)-f(a)}{b-a}$.
• Instantaneous rate of change: The rate of change at a single point, equal to the value of the derivative there.
• Derivative: The function $f'(x)$ giving the slope of the tangent line to f at each x where the defining limit exists.
• Secant line: A line through two points on a curve; its slope is an average rate of change.
• Tangent line: The line that touches a curve at a point with slope equal to the derivative at that point.
• Differentiability: The property of having a derivative at a point, which requires continuity plus a well-defined, finite tangent slope.
• Corner: A point where a continuous graph changes direction sharply, so left and right difference-quotient limits disagree, as in $|x|$ at 0.
• Vertical tangent: A point where the tangent line is vertical and the derivative is undefined, as in $\sqrt[3]{x}$ at 0.
• Cusp: A sharp point where one-sided slopes head to opposite infinities; continuous but not differentiable.
• Power Rule: The rule $\frac{d}{dx}x^r = rx^{r-1}$ for differentiating power functions.
• Product Rule: The rule $(fg)' = f'g + fg'$ for differentiating a product of two functions.
• Quotient Rule: The rule $\left(\frac{f}{g}\right)' = \frac{gf'-fg'}{g^2}$ for differentiating a ratio of two functions.
• Normal line: The line perpendicular to the tangent at a point, with slope $-\frac{1}{f'(a)}$.

Common mix-ups

• Continuity vs. differentiability runs one direction only. Differentiable means continuous, but $f(x) = |x|$ is continuous at 0 and not differentiable there. Exam questions love testing the false converse.
• $(fg)' \neq f'g'$ and $\left(\frac{f}{g}\right)' \neq \frac{f'}{g'}$. Products and quotients have their own rules. Test it yourself with $x \cdot x$ and you'll see the shortcut gives the wrong answer.
• In the Quotient Rule, the numerator is $gf' - fg'$ in that order. Flipping it negates your whole answer.
• Chain rule, implicit differentiation, related rates, and optimization are NOT in this unit. They live in Units 3, 4, and 5. Unit 2 is the definition plus the basic rules, so focus your review accordingly.