The derivative f'(x) is the limit of the difference quotient as h approaches 0. Unit 2 gives you the rules to compute that limit efficiently without returning to the definition every time: the Power Rule, constant and sum rules, Product Rule, Quotient Rule, and the standard derivatives of sin x, cos x, e^x, ln x, tan x, cot x, sec x, and csc x.
Fiveable's AP Calculus AB/BC Unit 2 review covers 10 topics, with guided practice questions for the unit.
Introduces the difference quotient (f(b)-f(a))/(b-a) as the average rate of change and the limit of that quotient as the instantaneous rate of change, connecting secant line slope to tangent line slope.
Extends the derivative to a function f'(x) defined by the limit of the difference quotient, introduces f'(x), y', and dy/dx notation, and connects the derivative to the slope of the tangent line and its equation.
Covers estimating f'(a) from tables using symmetric or one-sided difference quotients and from graphs by reading tangent line slope, including interpretation with units.
Establishes that differentiability implies continuity, identifies the four failure cases (corners, cusps, vertical tangents, discontinuities), and uses |x| and x^(1/3) as canonical examples.
Introduces d/dx(x^r) = r*x^(r-1) for all real r, including negative and fractional exponents, and emphasizes rewriting radicals and denominators as powers before differentiating.
Combines the constant rule, sum rule, difference rule, and constant multiple rule with the Power Rule to differentiate any polynomial term by term.
Presents the four essential non-polynomial derivatives and introduces the skill of recognizing a limit expression as a derivative definition to evaluate it using a known result.
Covers d/dx(f*g) = f'g + fg' with worked examples involving polynomial-trig and exponential-polynomial products, and addresses the common error of multiplying derivatives directly.
Covers d/dx(f/g) = (gf' - fg')/g^2, emphasizes the fixed subtraction order, and applies the rule to rational functions and trig ratios.
Derives d/dx(tan x) = sec^2 x, d/dx(cot x) = -csc^2 x, d/dx(sec x) = sec xtan x, and d/dx(csc x) = -csc xcot x by rewriting each as a ratio and applying the Quotient Rule.
Try AP-style questions on unit 2, then use the review notes below for anything you miss.
The average rate of change of f on [a, b] is (f(b) - f(a))/(b - a), which equals the slope of the secant line through those two points. The instantaneous rate of change at x = a is the limit of that quotient as the interval collapses: f'(a) = lim(h to 0) (f(a+h) - f(a))/h. The derivative function f'(x) extends this to every point in the domain. The slope of the tangent line at x = a equals f'(a), and the tangent line equation uses point-slope form: y - f(a) = f'(a)(x - a).
Given f(x) = x^2, use the limit definition to show f'(3) = 6, then write the equation of the tangent line at x = 3.
| Concept | Formula | Geometric meaning |
|---|---|---|
| Average rate of change | (f(b)-f(a))/(b-a) | Slope of secant line |
| Instantaneous rate of change | lim(h to 0)(f(a+h)-f(a))/h | Slope of tangent line |
| Derivative function | lim(h to 0)(f(x+h)-f(x))/h | Slope function for all x |
When you cannot compute a derivative algebraically, estimate it using nearby data. From a table, use the symmetric difference quotient (f(a+h) - f(a-h))/(2h) when interior values are available, or a one-sided quotient at endpoints. From a graph, estimate the slope of the tangent line by reading rise over run between two close points. Always include units when the context provides them.
A table gives f(2) = 5, f(3) = 8, f(4) = 13. Estimate f'(3) using the symmetric difference quotient.
| Method | Formula | Best used when |
|---|---|---|
| Symmetric difference quotient | (f(a+h)-f(a-h))/(2h) | Interior table point with equal spacing |
| Forward difference quotient | (f(a+h)-f(a))/h | Left endpoint or only right neighbor available |
| Backward difference quotient | (f(a)-f(a-h))/h | Right endpoint or only left neighbor available |
Differentiability implies continuity: if f is differentiable at x = a, it must be continuous there. The converse is false. A function can be continuous at a point but still fail to be differentiable. The four main failure cases are corners, cusps, vertical tangents, and discontinuities. To justify non-differentiability on the AP exam, name the specific reason rather than just identifying the point.
Explain why f(x) = |x - 2| is continuous but not differentiable at x = 2. Compute the left-hand and right-hand difference quotient limits to support your answer.
| Failure type | Example | One-sided limits of difference quotient |
|---|---|---|
| Corner | |x| at x=0 | Left: -1, Right: +1 (unequal, both finite) |
| Vertical tangent | x^(1/3) at x=0 | Both approach infinity |
| Discontinuity | Piecewise jump at x=a | At least one limit does not match function value |
The Power Rule states d/dx(x^r) = r*x^(r-1) for any real number r. It applies to integer, negative, and fractional exponents, so rewrite radicals and denominators as powers before differentiating. The constant rule, sum rule, difference rule, and constant multiple rule let you differentiate any polynomial term by term. The derivative of a constant is zero; a constant factor pulls out front.
Find the derivative of f(x) = 3x^4 - 5x^(1/2) + 7x^(-1) + 9 using the Power Rule and algebraic rules.
| Rule | Formula | Example |
|---|---|---|
| Power Rule | d/dx(x^r) = r*x^(r-1) | d/dx(x^5) = 5x^4 |
| Constant Multiple | d/dx(c*f) = c*f' | d/dx(4x^3) = 12x^2 |
| Sum/Difference | d/dx(f +/- g) = f' +/- g' | d/dx(x^2 + x) = 2x + 1 |
| Constant | d/dx(c) = 0 | d/dx(7) = 0 |
Four essential derivatives must be memorized: d/dx(sin x) = cos x, d/dx(cos x) = -sin x, d/dx(e^x) = e^x, and d/dx(ln x) = 1/x. The negative sign on the cosine derivative is the most common source of errors. The self-derivative property of e^x is unique among functions. These can be combined with sum, difference, and constant multiple rules. A secondary skill from this topic is recognizing a limit expression as a derivative definition and evaluating it using a known derivative.
Evaluate lim(h to 0) (sin(pi/6 + h) - sin(pi/6))/h by recognizing it as a derivative, then confirm by differentiating sin x directly.
| Function | Derivative | Key note |
|---|---|---|
| sin x | cos x | Positive; use radian measure |
| cos x | -sin x | Negative sign required |
| e^x | e^x | Self-derivative; unique to e |
| ln x | 1/x | Domain x > 0 |
When two differentiable functions are multiplied, use the Product Rule: d/dx(fg) = f'g + fg'. When one differentiable function is divided by another, use the Quotient Rule: d/dx(f/g) = (gf' - f*g')/g^2. The subtraction order in the Quotient Rule is fixed; reversing it changes the sign of the answer. Neither rule allows you to simply multiply or divide the individual derivatives.
Differentiate h(x) = x^2 * sin x using the Product Rule, then differentiate k(x) = e^x / (x^2 + 1) using the Quotient Rule.
| Rule | Formula | Common error |
|---|---|---|
| Product Rule | (fg)' = f'g + fg' | Multiplying f' and g' instead of using the formula |
| Quotient Rule | (f/g)' = (gf' - fg')/g^2 | Reversing subtraction order or forgetting to square the denominator |
The four remaining trig derivatives follow from rewriting each function as a ratio of sin x and cos x and applying the Quotient Rule. The results are: d/dx(tan x) = sec^2 x, d/dx(cot x) = -csc^2 x, d/dx(sec x) = sec x * tan x, d/dx(csc x) = -csc x * cot x. The negative signs on cot x and csc x derivatives are the most frequently missed details. All four require radian measure.
Derive d/dx(tan x) = sec^2 x by applying the Quotient Rule to sin x / cos x. Then state d/dx(sec x) and d/dx(csc x) from memory.
| Function | Rewrite | Derivative |
|---|---|---|
| tan x | sin x / cos x | sec^2 x |
| cot x | cos x / sin x | -csc^2 x |
| sec x | 1 / cos x | sec x * tan x |
| csc x | 1 / sin x | -csc x * cot x |
| Term | Definition |
|---|---|
| Average Rate of Change | The slope of the secant line between two points on f, computed as (f(b)-f(a))/(b-a) over the interval [a, b]. |
| Difference Quotient | The expression (f(a+h)-f(a))/h that measures average rate of change; its limit as h approaches 0 defines the derivative. |
| Instantaneous Rate of Change | The derivative f'(a), equal to lim(h to 0)(f(a+h)-f(a))/h; represents the slope of the tangent line at x = a. |
| Secant Line | A line through two points on a curve whose slope equals the average rate of change of the function over that interval. |
| Tangent Line | The line through (a, f(a)) with slope f'(a); it represents the best linear approximation to the curve at that point. |
| Slope of the Tangent Line | The value of the derivative f'(a) at a specific point; it equals the instantaneous rate of change of f at x = a. |
| f(x) | Prime notation for the derivative function; gives the instantaneous rate of change of f at any x in its domain. |
| Differentiable | A function is differentiable at x = a if the limit definition of the derivative exists there; differentiability implies continuity but not vice versa. |
| Vertical Tangent Line | Occurs when the difference quotient limit is infinite at a point, such as for f(x) = x^(1/3) at x = 0; the derivative does not exist there. |
| Constant Multiple Rule | d/dx(c*f(x)) = c*f'(x); a constant factor is preserved when differentiating. |
| Sum Rule | d/dx(f(x) + g(x)) = f'(x) + g'(x); derivatives of a sum split term by term. |
| Difference Rule | d/dx(f(x) - g(x)) = f'(x) - g'(x); derivatives of a difference split term by term. |
| Exponential Function | For the base e, d/dx(e^x) = e^x; the exponential function e^x is its own derivative, a property unique to this base. |
Miss rates come from Fiveable multiple-choice practice.
d/dx(cos x) = -sin x, d/dx(cot x) = -csc^2 x, and d/dx(csc x) = -csc x*cot x all carry a negative sign. The sine and tangent derivatives are positive; the cosine, cotangent, and cosecant derivatives are negative. Mixing these up is one of the most frequent errors on trig differentiation problems.
d/dx(f*g) is not f'*g'. The correct formula is f'g + fg'. Always write both terms before simplifying, especially when one factor is a trig or exponential function.
The numerator of the Quotient Rule is gf' - fg', not fg' - gf'. Reversing the order changes the sign of the entire derivative. A mnemonic like 'low d-high minus high d-low over low squared' can help keep the order straight.
The Power Rule requires the form x^r. Rewrite sqrt(x) as x^(1/2) and 1/x^3 as x^(-3) before differentiating. Applying the rule directly to radical or fraction notation without rewriting leads to incorrect exponents.
Continuity at a point does not guarantee differentiability. f(x) = |x| is continuous everywhere but not differentiable at x = 0. Always check for corners, cusps, or vertical tangents before claiming a derivative exists.
A common multiple-choice task presents a limit expression such as lim(h to 0)(cos(pi/3 + h) - cos(pi/3))/h and asks for its value. The skill is recognizing the expression as f'(a) for a known function and evaluating it using the appropriate derivative formula rather than computing the limit from scratch. This tests whether you understand the definition of the derivative, not just the shortcut rules.
Free-response problems involving piecewise functions often ask whether the function is differentiable at a join point. A complete response requires checking continuity first, then showing that the left-hand and right-hand difference quotient limits are equal. Stating only that the function looks smooth is not sufficient; you must show the one-sided derivative limits and confirm they match.
Both multiple-choice and free-response problems frequently ask you to differentiate a function that requires combining two or more rules (for example, the Product Rule applied to a polynomial and a trig function), then evaluate the derivative at a specific point to write a tangent line equation. Showing organized step-by-step work and using correct notation throughout is expected in free-response scoring.
Open the individual guides for Unit 2 when you want a closer review of one topic.
Use AP-style practice after you review the notes so you can check what you understand.
Practice free-response reasoning and compare your answer with scoring guidance.
Watch past review streams filtered to Unit 2 when you want a video walkthrough.
Open the Fiveable one-page unit review, then explore visual cheatsheets for a quick refresher.
Estimate your broader AP score goal after you review the course and exam format.
The AP Calc Unit 2 progress check in AP Classroom includes both MCQ and FRQ parts drawn from all 10 topics in the unit. MCQ questions typically test the Power Rule, Product Rule, Quotient Rule, derivatives of sin x, cos x, e^x, and ln x, and identifying where derivatives do not exist. FRQ parts ask you to apply derivative rules and interpret instantaneous rates of change in context. For the progress check, focus especially on Topics 2.4 through 2.10, since those cover the computational rules most frequently tested. You can find matched practice problems at AP Calc Unit 2.
AP Calc Unit 2 FRQs most often ask you to find a derivative using the Power Rule, Product Rule, or Quotient Rule, or to interpret an instantaneous rate of change in a real-world context. To practice, work through problems that require you to write out full derivative notation, show each rule step clearly, and explain what a derivative value means in context, since partial credit depends on clear reasoning. Good practice steps:
1. Drill Topics 2.5-2.9 (the core rules) until the algebra is automatic.
2. Practice writing answers using both f'(x) and dy/dx notation (Topic 2.2).
3. Review differentiability conditions (Topic 2.4) since FRQs sometimes ask you to justify why a derivative does or does not exist. Find FRQ-style practice at AP Calc Unit 2.
The best place to find AP Calc Unit 2 practice questions, including multiple-choice and practice test problems, is AP Calc Unit 2. That page has resources covering all 10 topics, from the limit definition of the derivative through the Product Rule, Quotient Rule, and trig derivatives. For MCQ practice, focus on problems that mix the Power Rule with Product and Quotient Rule applications, since those combinations appear most often.
Start AP Calc Unit 2 by making sure you understand the limit definition of the derivative (Topic 2.2) before moving to the shortcut rules, since exam questions sometimes ask you to use the definition directly. Then build your rule fluency in order: Power Rule, then Constant and Sum rules, then Product and Quotient Rules, then trig and exponential derivatives. A practical study plan:
1. Memorize the derivatives of sin x, cos x, e^x, ln x, and the six trig functions (Topics 2.7 and 2.10) as a set.
2. Practice the Product and Quotient Rules (Topics 2.8-2.9) with messy expressions so the algebra doesn't slow you down on the exam.
3. Review Topic 2.4 on differentiability and continuity, since it's a common MCQ trap.
4. Do timed mixed practice sets that combine multiple rules in one problem. All study resources for this unit are at AP Calc Unit 2.