Chain Rule Implicit Differentiation Test Answer Key with Scoring – AP Calculus AB

Question 1 Chain rule evaluation (2 points)

f′(x) = 2xx² + 3. At x = 1: 2/4 = 1/2. Answer (B).

Scoring: 2 points for the correct choice; 1 point for a correct chain rule setup.

Question 2 Exponential composition (2 points)

h′(x) = (2x − 4)e^{x2 − 4x}. At x = 3: (2)e^{9 − 12} = 2e⁻³. Answer (B).

Scoring: 2 points for the correct choice; 1 point for the correct inner derivative.

Question 3 Chain and product from a table (3 points)

(a) h′(1) = f′(g(1))·g′(1) = f′(2)·3 = (−4)(3) = −12.

(b) k′(1) = f′(1)g(1) + f(1)g′(1) = 6·2 + 5·3 = 27.

(c) m′(1) = g′(g(1))·g′(1) = g′(2)·3 = 2·3 = 6.

Scoring: 1 point per part.

Question 4 Tangent line to an implicit curve (3 points)

(a) Differentiate: y² + 2xydydx + 2xy + x²dydx = 0. Solve: dydx = −(y² + 2xy)2xy + x².

(b) At (1, 2): −(4 + 4)/(4 + 1) = −8/5. The line is y − 2 = −8/5(x − 1).

Scoring: Part (a): 1 point for differentiating both terms correctly, 1 point for solving. Part (b): 1 point.

Question 5 Inverse function derivative (3 points)

(a) Since f(1) = 1 + 4 + 3 = 8, g(8) = 1. Also f′(x) = 3x² + 4, so f′(1) = 7 and g′(8) = 1/7.

(b) The derivative of arctan u is u′/(1 + u²): 2x1 + x⁴.

Scoring: Part (a): 1 point for locating the input 1, 1 point for the reciprocal. Part (b): 1 point.

Question 6 First and second derivatives (3 points)

(a) Product rule: f′(x) = 2xe^−x − x²e^−x = (2x − x²)e^−x.

(b) Differentiate again: (2 − 2x)e^−x − (2x − x²)e^−x = (x² − 4x + 2)e^−x.

Scoring: Part (a): 1 point. Part (b): 1 point for the product rule setup, 1 point for the simplification.

Question 7 Slope on an ellipse (2 points)

Differentiate: 2x + 8ydydx = 0, so dydx = −x4y. At (2, 2): −2/8 = −1/4.

Scoring: 1 point for differentiating implicitly, 1 point for the correct slope.

Question 8 Derivative of a squared sine (2 points)

Chain rule: 2 sin(3x)·cos(3x)·3 = 6 sin(3x) cos(3x). Since 2 sin θ cos θ = sin 2θ, this equals 3 sin(6x).

Scoring: 1 point for the chain rule, 1 point for the simplification.