Derivative Rules Test – AP Calculus AB Math (Printable PDF)

Question 1 Product rule evaluation (2 points)

If f(x) = x³e^x, what is f′(1)?

1. 3e
2. 4e
3. e + 3
4. 4

Question 2 Quotient rule evaluation (2 points)

If f(x) = sin xx, what is f′(π)?

1. −1/π
2. 1/π
3. −π
4. 0

Question 3 Limit definition (3 points)

Use the limit definition of the derivative to find f′(5) for f(x) = 3x² − 2x. Show every step.

Question 4 Snow depth table (3 points)

The depth D(t) of snow, in centimeters, t hours after a storm begins is plotted.

1. Find the average rate of change of D on [0, 7]. (1 point)
2. Use the nearest data points to estimate D′(5). (1 point)
3. Interpret your answer to (b) with units. (1 point)

Question 5 Tangent line to a rational function (3 points)

Let f(x) = x² + 3x − 1.

1. Find f′(x). (2 points)
2. Write an equation of the tangent line at x = 2. (1 point)

Question 6 Piecewise differentiability (3 points)

Let f(x) = ax² + 1 for x ≤ 2, and f(x) = 3x + b for x > 2. The function is differentiable at x = 2.

1. Write the equation that makes f continuous at 2. (1 point)
2. Write the equation that makes the derivatives equal at 2. (1 point)
3. Find a and b. (1 point)

Question 7 Where the derivative fails (2 points)

The graph of f is shown on [−3, 8]. List every x-value in (−3, 8) where f′(x) does not exist, and give the reason for each.

Question 8 Trigonometric derivatives (2 points)

Find f′(x) for f(x) = 4 tan x − csc x.