Chain Rule Implicit Differentiation Worksheet with Answers – AP Calculus AB (Free PDF)

Problem 1 Chain rule with a power ★★★

Differentiate f(x) = (6x + 1)⁴ with respect to x.

Problem 2 Chain rule with a root ★★★

Compute the derivative of f(x) = √(3x² + 16).

Problem 3 Chain rule with e ★★★

Find f′(x) for f(x) = e^{x2 − 3x}.

Problem 4 Chain rule with a logarithm ★★★

Find f′(x) for f(x) = ln(x² + 4x).

Problem 5 Chain rule with sine ★★★

Find f′(x) for f(x) = sin(3x²).

Problem 6 Chain rule at a point ★★★

Let f(x) = (2x² − 1)³. Evaluate f′(1).

Problem 7 Chain rule from a table ★★★

Let h(x) = f(g(x)). Use the table to find h′(1).

Problem 8 Product rule with the chain rule ★★★

Use the product rule together with the chain rule to differentiate f(x) = x³e^−2x.

Problem 9 Quotient rule with the chain rule ★★★

Find f′(x) for f(x) = sin 2xx².

Problem 10 Slope on a circle ★★★

The circle x² + y² = 169 contains the point (5, 12). Find dydx and the slope of the tangent line at (5, 12).

Problem 11 Implicit differentiation with a product ★★★

The curve x²y + y³ = 10 contains (1, 2). Find dydx at that point.

Problem 12 Tangent line to an implicit curve ★★★

Write the equation of the line tangent to x² + xy + y² = 7 at (2, 1).

Problem 13 Derivative of an inverse function ★★★

Let f(x) = x³ + x, which is one-to-one, and let g be its inverse. Find g′(10).

Problem 14 Inverse derivative from a table ★★★

The function f is differentiable and one-to-one with the values shown. If g is its inverse, find g′(7).

Problem 15 Inverse trigonometric derivatives ★★★

Find the derivative of

1. arctan(4x)
2. arcsin(x²)

Problem 16 Second derivative of a sum ★★★

Let f(x) = x⁵ − 4x³ + e^2x. Find f″(x) and f″(0).

Problem 17 Second derivative of an implicit curve ★★★

On the circle x² + y² = 36, find d²ydx² at the point (0, 6).

Problem 18 Pressure on a climbing balloon ★★★

Air pressure, in kilopascals, at height h kilometers is P(h) = 100e^−0.2h. A balloon’s height is h(t) = 5 + 3t km after t hours. Find the rate of change of pressure at t = 2.

Problem 19 Camera tracking a rocket ★★★

A camera sits 20 feet from the base of a launch pad and tracks a rocket at height x feet. The angle of elevation is θ = arctan(x/20). Find dθdx when x = 15.

Problem 20 Horizontal and vertical tangents ★★★

Find all points on x² − xy + y² = 3 where the tangent line is horizontal and all points where it is vertical.