---
title: "AP Calc AB/BC Unit 2 Review: Fundamentals of Differentiation"
description: "AP Calculus AB/BC Unit 2 covers Applying the Power Rule and Derivatives of cos x, sinx, e^x, and ln x. Study guides, practice questions, and key terms."
canonical: "https://fiveable.me/ap-calc/unit-2"
type: "unit"
subject: "AP Calculus AB/BC"
unit: "Unit 2 – Fundamentals of Differentiation"
---

# AP Calc AB/BC Unit 2 Review: Fundamentals of Differentiation

## Overview

Unit 2 formalizes the derivative as a limit of a difference quotient, then builds a toolkit of rules for computing derivatives efficiently. You will move from estimating and defining derivatives to applying the Power Rule, basic algebraic rules, and the derivatives of sin x, cos x, e^x, ln x, and the four remaining trig functions, plus the Product and Quotient Rules.

## AP CED Alignment

This unit hub is organized around AP Course and Exam Description topics, skills, and exam task types when they are available in the source data.
- 2.1: Defining Average and Instantaneous Rates of Change at a Point
- 2.2: Defining the Derivative of a Function and Using Derivative Notation
- 2.3: Estimating Derivatives of a Function at a Point
- 2.4: Connecting Differentiability and Continuity
- 2.5: Applying the Power Rule
- 2.6: Derivative Rules: Constant, Sum, Difference, and Constant Multiple
- 2.7: Derivatives of cos x, sin x, e^x, and ln x
- 2.8: The Product Rule
- 2.9: The Quotient Rule
- 2.10: Derivatives of Tangent, Cotangent, Secant, and Cosecant Functions
- 2.1-2.2: Defining the Derivative from Average to Instantaneous Rate of Change
- 2.3: Estimating Derivatives from Tables and Graphs
- 2.4: Differentiability and Continuity
- 2.5-2.6: Power Rule and Algebraic Derivative Rules
- 2.7: Derivatives of sin x, cos x, e^x, and ln x
- 2.8-2.9: Product Rule and Quotient Rule
- 2.10: Derivatives of tan x, cot x, sec x, and csc x
- Practice 1 - Implementing Mathematical Processes
- Practice 3 - Justification
- FRQs – Graphing calculator required
- FRQs – No graphing calculator

## Topics

- [2.1: Defining Average and Instantaneous Rates of Change at a Point](/ap-calc/unit-2/defining-average-instantaneous-rates-change-at-point/study-guide/5ozgLc7Ucg4El3O0fV28): 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.
- [2.2: Defining the Derivative of a Function and Using Derivative Notation](/ap-calc/unit-2/defining-derivative-function-using-derivative-notation/study-guide/j9KaEWbB5OECijykhxCD): 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.
- [2.3: Estimating Derivatives of a Function at a Point](/ap-calc/unit-2/estimating-derivatives-function-at-point/study-guide/EzDxIj2eOHj3Ysb68Ew5): 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.
- [2.4: Connecting Differentiability and Continuity](/ap-calc/unit-2/determining-when-derivatives-do-do-not-exist/study-guide/Lk6aXtIExtqciduDNGhk): 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.
- [2.5: Applying the Power Rule](/ap-calc/unit-2/applying-power-rule/study-guide/GMr6EEbZezsP1DvqrpEk): 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.
- [2.6: Derivative Rules: Constant, Sum, Difference, and Constant Multiple](/ap-calc/unit-2/derivative-rules-constant-sum-difference-constant-multiple/study-guide/0DS7QaXV5BZFYbNZySlm): Combines the constant rule, sum rule, difference rule, and constant multiple rule with the Power Rule to differentiate any polynomial term by term.
- [2.7: Derivatives of cos x, sin x, e^x, and ln x](/ap-calc/unit-2/derivatives-cos-x-sinx-ex-ln-x/study-guide/SbmDK3t2kYI0u2wLz9Hv): 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.
- [2.8: The Product Rule](/ap-calc/unit-2/product-rule/study-guide/qQXYTmpHvjsqAWOVzcEw): 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.
- [2.9: The Quotient Rule](/ap-calc/unit-2/quotient-rule/study-guide/qjB4zQXXy6ps14FlOXLc): Covers d/dx(f/g) = (gf' - fg')/g^2, emphasizes the fixed subtraction order, and applies the rule to rational functions and trig ratios.
- [2.10: Derivatives of Tangent, Cotangent, Secant, and Cosecant Functions](/ap-calc/unit-2/derivatives-tangent-cotangent-secant-cosecant-functions/study-guide/wkBAhxiFZ1wV1M5mwLwt): Derives d/dx(tan x) = sec^2 x, d/dx(cot x) = -csc^2 x, d/dx(sec x) = sec x*tan x, and d/dx(csc x) = -csc x*cot x by rewriting each as a ratio and applying the Quotient Rule.

## Hardest Topics And Analytics

Snapshot: practice snapshot
This snapshot uses Fiveable practice activity to show where students tend to miss questions and which review moves are worth prioritizing first.
- **60% average MCQ accuracy** (Across 5.9k multiple-choice practice attempts for this unit.)
- **5.9k MCQ attempts** (Practice activity included in this snapshot.)
- **69% average FRQ score** (Across 23 scored free-response attempts for this unit.)
- **2.2: Defining the Derivative of a Function and Using Derivative Notation**: 47% MCQ miss rate across 1085 attempts. Review Defining the Derivative of a Function and Using Derivative Notation with attention to how the concept appears in AP-style source and evidence questions.
- **2.6: Derivative Rules: Constant, Sum, Difference, and Constant Multiple**: 30% MCQ miss rate across 519 attempts. Review Derivative Rules: Constant, Sum, Difference, and Constant Multiple with attention to how the concept appears in AP-style source and evidence questions.

## Review Notes

### 2.1-2.2: Defining the Derivative from Average to Instantaneous Rate of Change

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).

- **Average rate of change**: (f(b) - f(a))/(b - a); slope of the secant line between (a, f(a)) and (b, f(b)).
- **Limit definition of the derivative**: f'(a) = lim(h to 0) (f(a+h) - f(a))/h, or equivalently lim(x to a) (f(x) - f(a))/(x - a).
- **Derivative function**: f'(x) = lim(h to 0) (f(x+h) - f(x))/h; gives the instantaneous rate of change at every x in the domain.
- **Derivative notation**: f'(x), y', and dy/dx all represent the same derivative; choose notation to match the problem context.
- **Tangent line**: The line through (a, f(a)) with slope f'(a); equation is y - f(a) = f'(a)(x - a).

**Checkpoint:** 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

### 2.3: Estimating Derivatives from Tables and Graphs

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.

- **Symmetric difference quotient**: (f(a+h) - f(a-h))/(2h); generally the most accurate table-based estimate for f'(a) at interior points.
- **One-sided estimate**: Use (f(a+h) - f(a))/h or (f(a) - f(a-h))/h when only one neighboring value is available, such as at a table endpoint.
- **Graphical estimation**: Read two nearby points on the curve and compute slope; the closer the points, the better the approximation of the tangent slope.

**Checkpoint:** 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

### 2.4: Differentiability and Continuity

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.

- **Differentiable implies continuous**: If f'(a) exists, then f is continuous at a. Contrapositive: if f is discontinuous at a, then f is not differentiable at a.
- **Corner**: Left-hand and right-hand difference quotient limits both exist but are not equal; example: f(x) = |x| at x = 0.
- **Vertical tangent**: The difference quotient limit is infinite; example: f(x) = x^(1/3) at x = 0, where the slope grows without bound.
- **Cusp**: One-sided difference quotient limits are infinite with opposite signs; the graph has a sharp point with infinite steepness.
- **Discontinuity**: Any jump, removable, or infinite discontinuity at a point prevents differentiability there.

**Checkpoint:** 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

### 2.5-2.6: Power Rule and Algebraic Derivative Rules

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.

- **Power Rule**: d/dx(x^r) = r*x^(r-1); works for all real r, including r = 1/2 (square root) and r = -1 (reciprocal).
- **Constant rule**: d/dx(c) = 0 for any constant c.
- **Constant multiple rule**: d/dx(c*f(x)) = c*f'(x); the constant factor is preserved.
- **Sum and difference rules**: d/dx(f(x) +/- g(x)) = f'(x) +/- g'(x); differentiate term by term.
- **Rewriting before differentiating**: Convert sqrt(x) to x^(1/2) and 1/x^2 to x^(-2) before applying the Power Rule to avoid errors.

**Checkpoint:** 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

### 2.7: Derivatives of sin x, cos x, e^x, and ln x

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.

- **d/dx(sin x) = cos x**: Derived from the limit definition using the sine addition formula and the special limits lim(sin h/h) = 1 and lim((1-cos h)/h) = 0.
- **d/dx(cos x) = -sin x**: The negative sign is essential; forgetting it is one of the most common errors on trig derivative problems.
- **d/dx(e^x) = e^x**: The exponential function e^x is its own derivative; this property defines e and makes it central to differential equations.
- **d/dx(ln x) = 1/x**: Valid for x > 0; derived using the inverse function relationship between e^x and ln x.
- **Limit-as-derivative recognition**: If a limit matches the form lim(h to 0)(f(a+h)-f(a))/h, evaluate it as f'(a) using a known derivative instead of computing the limit directly.

**Checkpoint:** 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

### 2.8-2.9: Product Rule and Quotient Rule

When two differentiable functions are multiplied, use the Product Rule: d/dx(f*g) = f'*g + f*g'. When one differentiable function is divided by another, use the Quotient Rule: d/dx(f/g) = (g*f' - 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.

- **Product Rule**: d/dx(f*g) = f'*g + f*g'; first times derivative of second, plus second times derivative of first.
- **Quotient Rule**: d/dx(f/g) = (g*f' - f*g')/g^2; denominator times derivative of numerator minus numerator times derivative of denominator, all over denominator squared.
- **Order matters in Quotient Rule**: g*f' - f*g' is not the same as f*g' - g*f'; always put the denominator function first in the numerator of the formula.
- **When to use each rule**: Product Rule for expressions written as a product; Quotient Rule for explicit fractions. Rewriting a quotient as a product with a negative exponent and using the Product Rule is an alternative.

**Checkpoint:** 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

### 2.10: Derivatives of tan x, cot x, sec x, and csc x

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.

- **d/dx(tan x) = sec^2 x**: Derived by applying the Quotient Rule to sin x / cos x; uses the identity sin^2 x + cos^2 x = 1 to simplify.
- **d/dx(cot x) = -csc^2 x**: Derived from cos x / sin x via the Quotient Rule; the negative sign is required.
- **d/dx(sec x) = sec x * tan x**: Derived from 1/cos x using the reciprocal or Quotient Rule.
- **d/dx(csc x) = -csc x * cot x**: Derived from 1/sin x; the negative sign is required and often forgotten.
- **Derivation strategy**: Rewrite tan x = sin x/cos x, cot x = cos x/sin x, sec x = 1/cos x, csc x = 1/sin x, then apply the Quotient Rule to derive each formula rather than memorizing without understanding.

**Checkpoint:** 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

## Study Guides

- [2.6 Derivative Rules: Constant, Sum, Difference, and Constant Multiple](/ap-calc/unit-2/derivative-rules-constant-sum-difference-constant-multiple/study-guide/0DS7QaXV5BZFYbNZySlm)
- [2.1 Defining Average and Instantaneous Rates of Change at a Point](/ap-calc/unit-2/defining-average-instantaneous-rates-change-at-point/study-guide/5ozgLc7Ucg4El3O0fV28)
- [2.3 Estimating Derivatives of a Function at a Point](/ap-calc/unit-2/estimating-derivatives-function-at-point/study-guide/EzDxIj2eOHj3Ysb68Ew5)
- [2.5 Applying the Power Rule](/ap-calc/unit-2/applying-power-rule/study-guide/GMr6EEbZezsP1DvqrpEk)
- [2.4 Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist](/ap-calc/unit-2/determining-when-derivatives-do-do-not-exist/study-guide/Lk6aXtIExtqciduDNGhk)
- [2.7 Derivatives of cos x, sinx, e^x, and ln x](/ap-calc/unit-2/derivatives-cos-x-sinx-ex-ln-x/study-guide/SbmDK3t2kYI0u2wLz9Hv)
- [2.2 Defining the Derivative of a Function and Using Derivative Notation](/ap-calc/unit-2/defining-derivative-function-using-derivative-notation/study-guide/j9KaEWbB5OECijykhxCD)
- [2.8 The Product Rule](/ap-calc/unit-2/product-rule/study-guide/qQXYTmpHvjsqAWOVzcEw)
- [2.9 The Quotient Rule](/ap-calc/unit-2/quotient-rule/study-guide/qjB4zQXXy6ps14FlOXLc)
- [2.10 Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions](/ap-calc/unit-2/derivatives-tangent-cotangent-secant-cosecant-functions/study-guide/wkBAhxiFZ1wV1M5mwLwt)

## Practice Preview

### Multiple-choice practice

- **AP-style practice question**: Practice 1 - Implementing Mathematical Processes | The function $$h(x) = \frac{e^x}{x^2 + 4}$$ has derivative $$h'(x) = \frac{(x^2+4)e^x - 2xe^x}{(x^2+4)^2}$$ by the quotient rule. A student constructs the second-degree Taylor polynomial $$P_2(x)$$ centered at $$x = 0$$ to approximate $$h(0.4)$$. It is known that $$h(0) = 0.25$$, $$h'(0) = 0.25$$, $$h''(0) = -0.1875$$, and $$|h'''(x)| \leq 1.2$$ for all $$x \in [0, 0.4]$$. Which statement correctly describes how the error bound constrains the actual value of $$h(0.4)$$?
- **AP-style practice question**: Practice 3 - Justification | Consider $$g(x) = 4(3x^3 - 2x)$$. Which rule should be applied first to find $$g'(x)$$?
- **AP-style practice question**: Practice 3 - Justification | To find $$\frac{d}{dx}[3(x^5 - 2x^3)]$$, which rule should be applied first, and what is the justification?
- **AP-style practice question**: Practice 3 - Justification | A function is defined as $$f(x) = \frac{|x|}{x^2 + 3}$$ for all real $$x$$. Which hypothesis of the Quotient Rule fails at $$x = 0$$?
- **AP-style practice question**: Practice 3 - Justification | A student applies the Power Rule to $$f(x) = x^{-2}$$ on the domain $$x \ne 0$$ and obtains $$f'(x) = -2x^{-3}$$. Which condition confirms this application is valid?
- **AP-style practice question**: Practice 3 - Justification | A student applies the Power Rule to $$s(x) = 3x^{-1/2}$$ on $$x > 0$$ and obtains $$s'(x) = -\frac{3}{2}x^{-3/2}$$. Which statement correctly confirms the Power Rule hypothesis is satisfied?

### FRQ practice

- **Coffee temperature cooling over time**: FRQs – Graphing calculator required | Coffee temperature cooling over time
- **Bacterial population growth modeled by exponential function**: FRQs – No graphing calculator | Bacterial population growth modeled by exponential function

## Key Terms

- **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.

## Common Mistakes

- **Forgetting the negative sign on d/dx(cos x) and d/dx(cot x) and d/dx(csc x)**: 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.
- **Multiplying derivatives instead of using the Product Rule**: d/dx(f*g) is not f'*g'. The correct formula is f'*g + f*g'. Always write both terms before simplifying, especially when one factor is a trig or exponential function.
- **Reversing the subtraction order in the Quotient Rule**: The numerator of the Quotient Rule is g*f' - f*g', not f*g' - g*f'. 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.
- **Applying the Power Rule to expressions not yet in power form**: 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.
- **Concluding that a continuous function is differentiable**: 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.

## Exam Connections

- **Limit-as-derivative recognition in multiple choice**: 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.
- **Justifying differentiability in free response**: 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.
- **Multi-rule differentiation and tangent line equations**: 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.

## Final Review Checklist

- **Unit 2 final review checklist**: Use this list to confirm you can handle every major skill before the exam.
- **Write and evaluate the limit definition of the derivative**: Set up lim(h to 0)(f(a+h)-f(a))/h for a given function, simplify algebraically, and evaluate the limit to find f'(a).
- **Estimate derivatives from tables and graphs**: Apply the symmetric difference quotient at interior table points, use one-sided quotients at endpoints, and read tangent slope from a graph with correct units.
- **Identify non-differentiable points and explain why**: Recognize corners, cusps, vertical tangents, and discontinuities and state the specific reason the derivative does not exist at each.
- **Differentiate polynomials and power functions fluently**: Apply the Power Rule with the constant, sum, difference, and constant multiple rules to any polynomial or expression with integer, negative, or fractional exponents.
- **Recall and apply the four basic non-polynomial derivatives**: Produce d/dx(sin x), d/dx(cos x), d/dx(e^x), and d/dx(ln x) from memory and combine them with algebraic rules.
- **Apply the Product Rule and Quotient Rule correctly**: Identify when each rule is needed, write both terms of the Product Rule before simplifying, and keep the subtraction order correct in the Quotient Rule numerator.
- **State and derive the four remaining trig derivatives**: Produce d/dx(tan x), d/dx(cot x), d/dx(sec x), and d/dx(csc x), including correct negative signs, and be able to derive each from the Quotient Rule.

## Study Plan

- **Step 1: Solidify the limit definition and notation (Topics 2.1-2.2)**: Work through the difference quotient for at least three functions (a polynomial, a square root, and a trig function) using both the h-to-0 and x-to-a forms. Practice writing the tangent line equation at a specified point. Review the topic guides for 2.1 and 2.2 and check that you can switch fluently between f'(x), y', and dy/dx notation.
- **Step 2: Practice estimation and differentiability (Topics 2.3-2.4)**: Work several table-based estimation problems using symmetric and one-sided difference quotients, including problems that give units. Then sketch or analyze piecewise and absolute value functions to identify non-differentiable points and write a one-sentence justification for each. Use the topic guides for 2.3 and 2.4 to check your reasoning.
- **Step 3: Build fluency with Power Rule and algebraic rules (Topics 2.5-2.6)**: Differentiate at least ten polynomial and power function expressions, including ones with negative and fractional exponents. Rewrite radical and fraction forms before applying the Power Rule. Time yourself to build speed, since these rules appear in nearly every later problem.
- **Step 4: Memorize and apply the core non-polynomial derivatives (Topic 2.7)**: Write the four derivatives (sin x, cos x, e^x, ln x) from memory, then practice combining them with sum and constant multiple rules. Work two or three limit problems where you recognize the limit as a derivative definition and evaluate it using a known result rather than computing the limit directly.
- **Step 5: Practice Product Rule, Quotient Rule, and trig derivatives (Topics 2.8-2.10)**: Differentiate expressions that require the Product Rule, the Quotient Rule, and the four remaining trig derivatives. Focus on sign accuracy in the Quotient Rule numerator and on the negative signs in d/dx(cot x) and d/dx(csc x). Use the available FRQ practice to work multi-step problems that combine several rules in one expression.

## More Ways To Review

- [Topic study guides](/ap-calc/unit-2#topics)
- [FRQ practice](/ap-calc/frq-practice)
- [Cram archive videos](/cram-archives?subject=ap-calculus&unit=unit-2)
- [Key terms](/ap-calc/key-terms)

## FAQs

### What topics are covered in AP Calc Unit 2?

AP Calc Unit 2 covers 10 topics on the definition and fundamental properties of differentiation. You'll work through average and instantaneous rates of change, the limit definition of the derivative, derivative notation, differentiability vs. continuity, the Power Rule, the Product Rule, the Quotient Rule, and derivatives of sin x, cos x, e^x, ln x, tan x, cot x, sec x, and csc x. Here's the full topic list:
- 2.1 Average and Instantaneous Rates of Change at a Point
- 2.2 Defining the Derivative and Using Derivative Notation
- 2.3 Estimating Derivatives at a Point
- 2.4 Differentiability and Continuity
- 2.5 The Power Rule
- 2.6 Constant, Sum, Difference, and Constant Multiple Rules
- 2.7 Derivatives of cos x, sin x, e^x, and ln x
- 2.8 The Product Rule
- 2.9 The Quotient Rule
- 2.10 Derivatives of tan x, cot x, sec x, and csc x See practice and study resources at [AP Calc Unit 2](/ap-calc/unit-2).

### How much of the AP Calc exam is Unit 2?

AP Calc Unit 2 makes up 10-12% of the AP exam, making it one of the more heavily tested foundational units. It covers the definition of the derivative, the Power Rule, the Product Rule, the Quotient Rule, and derivatives of trigonometric and exponential functions. A solid grasp of these rules also supports nearly every other unit on the exam.

### What's on the AP Calc Unit 2 progress check (MCQ and FRQ)?

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).

### How do I practice AP Calc Unit 2 FRQs?

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](/ap-calc/unit-2).

### Where can I find AP Calc Unit 2 practice questions?

The best place to find AP Calc Unit 2 practice questions, including multiple-choice and practice test problems, is [AP Calc Unit 2](/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.

### How should I study AP Calc Unit 2?

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](/ap-calc/unit-2).

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Review Topic 2.4 on differentiability and continuity, since it's a common MCQ trap.\n4. Do timed mixed practice sets that combine multiple rules in one problem. All study resources for this unit are at <a href=\"/ap-calc/unit-2\">AP Calc Unit 2</a>."}}]}
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