AP Calc AB Differentiation Definition and Rules

Question 1

If f(x) = x² sin x, what is f′(π)?

Accepted Answer

−π²

Answer

By the product rule, f′(x) = 2x sin x + x² cos x. At x = π this is 2π(0) + π²(−1) = −π².

Question 2

Let f(x) = x³. What is the value of lim_{h→0} [(2 + h)³ − 8] / h?

Accepted Answer

12

Answer

The limit is the definition of f′(2) for f(x) = x³. Since f′(x) = 3x², f′(2) = 12.

Question 3

What is d/dx of √(3x² + 1)?

Accepted Answer

3x / √(3x² + 1)

Answer

By the chain rule, the derivative is (1/2)(3x² + 1)^(−1/2) · 6x = 3x / √(3x² + 1).

Question 4

The point (1, 2) lies on the curve x² + xy + y² = 7. What is dy/dx at that point?

Accepted Answer

−4/5

Answer

Differentiating implicitly gives 2x + y + x(dy/dx) + 2y(dy/dx) = 0. At (1, 2) this is 4 + 5(dy/dx) = 0, so dy/dx = −4/5.

Question 5

Let f(x) = x³ + x + 1, which is one-to-one. If g is the inverse function of f, what is g′(3)?

Accepted Answer

1/4

Answer

Since f(1) = 3, g(3) = 1. Then g′(3) = 1 / f′(1) = 1/(3·1² + 1) = 1/4.

AP Calc AB Practice Test