Differentiation Definition and Rules Flashcards

If f(x) = x² sin x, what is f′(π)?

Answer: −π²

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

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

Answer: 12

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

What is d/dx of √(3x² + 1)?

Answer: 3x / √(3x² + 1)

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

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

Answer: −4/5

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.

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

Answer: 1/4

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

If f(x) = x² / (x + 1), what is f′(1)?

Answer: 3/4

By the quotient rule, f′(x) = [2x(x + 1) − x²] / (x + 1)² = (x² + 2x) / (x + 1)². At x = 1 this is 3/4.

Which of the following functions is continuous at x = 0 but not differentiable at x = 0?

Answer: f(x) = x^(2/3)

For f(x) = x^(2/3), f′(x) = (2/3)x^(−1/3), which is unbounded near 0, so there is a cusp at x = 0. The other three functions are differentiable at 0.

What is d/dx [arctan(3x)]?

Answer: 3 / (1 + 9x²)

The derivative of arctan u is u′ / (1 + u²). With u = 3x, this gives 3 / (1 + 9x²).

What is d/dx [5^x]?

Answer: 5^x ln 5

For a constant base a > 0, d/dx [a^x] = a^x ln a. With a = 5, the derivative is 5^x ln 5.

Suppose f(2) = 1 and f′(2) = 4. If h(x) = [f(x)]³, what is h′(2)?

Answer: 12

By the chain rule, h′(x) = 3[f(x)]² · f′(x). At x = 2, h′(2) = 3(1)²(4) = 12.