AP Calculus AB: Product Rule, Quotient Rule, and Differentiability — Drill 1

About This Drill

AP Calculus AB: Product Rule, Quotient Rule, and Differentiability — Drill 1 is a Math practice drill covering Product Rule, Quotient Rule, and Differentiability. It contains 5 original questions created by Brian Stewart, a Barron's test prep author with over 20 years of tutoring experience.

Practice applying the product and quotient rules to differentiate functions, and explore the relationship between differentiability and continuity. This drill targets common errors including treating the derivative of a product as the product of derivatives and confusing non-differentiable points with discontinuities.

Questions in This Drill

1. Let \( f(x) = x^3 \sin x \). Find \( f'(x) \).
2. Let \( g(x) = \dfrac{e^x}{x^2} \). Find \( g'(x) \).
3. Which of the following statements is always true?
4. The function \( f \) is defined as \( f(x) = x^2 + 1 \) for \( x < 2 \) and \( f(x) = 3x - 1 \) for \( x \geq 2 \). Which of the following correctly describes \( f \) at \( x = 2 \)?
5. Suppose \( h(x) = f(x) \cdot g(x) \), where \( f(2) = 3 \), \( f'(2) = -1 \), \( g(2) = 4 \), and \( g'(2) = 5 \). What is \( h'(2) \)?