Chain Rule Implicit Differentiation Test – AP Calculus AB Math (Printable PDF)

Question 1 Chain rule evaluation (2 points)

If f(x) = ln(x² + 3), what is f′(1)?

1. 1/4
2. 1/2
3. 2
4. 1/3

Question 2 Exponential composition (2 points)

If h(x) = e^{x2 − 4x}, what is h′(3)?

1. e⁻³
2. 2e⁻³
3. 2
4. 6e⁻³

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

Values of f, g and their derivatives are given.

1. If h(x) = f(g(x)), find h′(1). (1 point)
2. If k(x) = f(x)g(x), find k′(1). (1 point)
3. If m(x) = g(g(x)), find m′(1). (1 point)

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

The curve xy² + x²y = 6 passes through (1, 2).

1. Find dydx in terms of x and y. (2 points)
2. Write an equation of the tangent line at (1, 2). (1 point)

Question 5 Inverse function derivative (3 points)

Let f(x) = x³ + 4x + 3, which is increasing, and let g be its inverse.

1. Find g′(8). (2 points)
2. Find the derivative of arctan(x²). (1 point)

Question 6 First and second derivatives (3 points)

Let f(x) = x²e^−x.

1. Find f′(x). (1 point)
2. Find f″(x). (2 points)

Question 7 Slope on an ellipse (2 points)

The ellipse x² + 4y² = 20 is shown. Find the slope of the tangent line at (2, 2).

Question 8 Derivative of a squared sine (2 points)

Find the derivative of f(x) = sin²(3x) and write it in the form a sin(bx).