Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
These 20 problems cover AP Calculus AB Unit 2: average and instantaneous rates of change, the limit definition, differentiability, and the rules for powers, products, quotients and trigonometric functions. The first problems are single-step; later ones need several rules at once.
The last six problems are multi-step contexts such as a drone flight and a drug concentration. Show the derivative first, then evaluate or solve.
Standards: AP Calculus AB Unit 2 · Level: Grade 12 AP Calculus AB
Problem 1 Average rate of change ★★★
Find the average rate of change of f(x) = x3 − 5x on the interval [1, 4].
Problem 2 Rate of change from a table ★★★
The elevation E (feet) of a hiking trail at mile m is shown. Find the average rate of change from mile 2 to mile 6 and give units.
| m (miles) | 0 | 2 | 4 | 6 |
|---|---|---|---|---|
| E(m) | 1,150 | 1,310 | 1,420 | 1,630 |
Problem 3 Power rule with a polynomial ★★★
Find f′(x) for f(x) = 7x4 − 5x2 + 9.
Problem 4 Power rule with roots and fractions ★★★
Find f′(x) for f(x) = 10√x − 12x4.
Problem 5 Derivatives of sine and cosine ★★★
Find f′(x) for f(x) = 5 sin x − 2 cos x.
Problem 6 Exponential and logarithm terms ★★★
Find f′(x) for f(x) = 3ex + 4 ln x − x.
Problem 7 Limit definition at a point ★★★
Apply the difference quotient with h → 0 to compute f′(5) when f(x) = x2 + 7x.
Problem 8 Product rule ★★★
Find f′(x) for f(x) = (3x2 + 1)(x3 − 4x).
Problem 9 Quotient rule ★★★
Find f′(x) for f(x) = 2x + 5x2 + 1.
Problem 10 Equation of a tangent line ★★★
Write an equation of the line tangent to f(x) = x3 − 3x + 1 at x = 2.
Problem 11 Derivatives of tangent and secant ★★★
Find f′(x) for f(x) = tan x − 2 sec x.
Problem 12 Product rule with a trig factor ★★★
Let f(x) = x2 sin x. Find f′(x) and evaluate f′(π).
Problem 13 Quotient rule with a logarithm ★★★
Let f(x) = ln xx2. Find f′(x) and evaluate f′(e).
Problem 14 Differentiable piecewise function ★★★
Let f(x) = x2 for x ≤ 3 and f(x) = ax + b for x > 3. Find a and b so that f is differentiable at x = 3.
Problem 15 Checking differentiability at a corner ★★★
Let f(x) = |x − 3|. Find the slope of the graph just to the left and just to the right of x = 3, and decide whether f′(3) exists.
Problem 16 Estimating a derivative from a table ★★★
The depth D(t) of a river at a gauge, in feet, is recorded. Use the points nearest t = 4 to estimate D′(4), with units.
| t (hours) | 2 | 4 | 6 | 9 |
|---|---|---|---|---|
| D(t) | 18 | 29 | 44 | 62 |
Problem 17 Horizontal tangent lines ★★★
Find every point on the graph of f(x) = x3 − 12x + 5 where the tangent line is horizontal.
Problem 18 Drone height ★★★
A drone’s height in feet is h(t) = −4t2 + 36t after t seconds. Find its average velocity on [1, 3], its velocity at t = 2, and the time when its velocity is zero.
Problem 19 Concentration of a medicine ★★★
The concentration of a drug in the blood is C(t) = 8tt2 + 4 mg/L, t hours after injection. Find C′(t), evaluate C′(1), and find when the concentration stops rising.
Problem 20 Tangent line through the origin ★★★
A line through the origin is tangent to the graph of f(x) = ln x. Find the point of tangency and the equation of the line.
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