← Back to the Derivative Rules test
Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
The worked solutions below carry a scoring guide. Give credit for correct setup even when a later arithmetic slip changes the final number, but require the correct rule to be named or clearly used.
Standards: AP Calculus AB Unit 2 · Level: Grade 12 AP Calculus AB
Question 1 Product rule evaluation (2 points)
f′(x) = 3x2ex + x3ex = (x3 + 3x2)ex. So f′(1) = 4e. Answer (B).
Scoring: 2 points for the correct choice; 1 point for a correct application of the product rule.
Question 2 Quotient rule evaluation (2 points)
f′(x) = x cos x − sin xx2. At x = π: (π(−1) − 0)/π2 = −1/π. Answer (A).
Scoring: 2 points for the correct choice; 1 point for a correct quotient rule setup.
Question 3 Limit definition (3 points)
f(5) = 75 − 10 = 65. The difference quotient is 3(5 + h)2 − 2(5 + h) − 65h = 28h + 3h2h = 28 + 3h. As h → 0: f′(5) = 28.
Scoring: 1 point for the correct difference quotient, 1 point for expanding and simplifying, 1 point for the limit.
Question 4 Snow depth table (3 points)
(a) (25 − 4)/(7 − 0) = 3 cm/h.
(b) Use the points at 3 and 7: (25 − 11)/(7 − 3) = 14/4 = 3.5 cm/h.
(c) At 5 hours after the storm began, the snow depth is increasing at about 3.5 centimeters per hour.
Scoring: 1 point per part; units required in (b) or (c).
Question 5 Tangent line to a rational function (3 points)
(a) Quotient rule: f′(x) = 2x(x − 1) − (x2 + 3)(x − 1)2 = x2 − 2x − 3(x − 1)2.
(b) f(2) = 7 and f′(2) = (4 − 4 − 3)/1 = −3. The line is y − 7 = −3(x − 2), or y = −3x + 13.
Scoring: Part (a): 1 point for the setup, 1 for the simplification. Part (b): 1 point.
Question 6 Piecewise differentiability (3 points)
(a) Left value 4a + 1 equals right value 6 + b.
(b) The left derivative is 2ax = 4a at x = 2, and the right derivative is 3. So 4a = 3.
(c) a = 3/4. Then 3 + 1 = 6 + b, so b = −2.
Scoring: 1 point per part.
Question 7 Where the derivative fails (2 points)
At x = 1 the graph has a corner, with slope −1 on the left and +1 on the right. At x = 3 the graph has a jump (the function is discontinuous), so it cannot be differentiable. x = 1 and x = 3.
Scoring: 1 point for each correct value with its reason.
Question 8 Trigonometric derivatives (2 points)
The derivative of tan x is sec2 x. The derivative of csc x is −csc x cot x. So f′(x) = 4 sec2 x + csc x cot x.
Scoring: 1 point per term.
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