Question 1

Which of the following is the correct derivative of the function $$f(x) = 7$$?

Accepted Answer

$$f'(x) = 0$$

Question 2

Using the power rule, what is the derivative of $$f(x) = x$$^{\frac{1}$${2}}$$?

Accepted Answer

$$f'(x) = \frac{1}{2}x$$^{-\frac{1}$${2}}$$

Question 3

Suppose $$h(x) = 4x^3$ - \frac{1}{\sqrt{x}}. $$What is $$h'(x)$$?

Accepted Answer

$$h'(x) = 12x^2$ - \frac{1}{2}x$$^{-\frac{3}$${2}}$$

Question 4

If $$g(x) = 5x^7$, what is g$$'(x)$$ using the constant multiple rule?

Accepted Answer

$$g'(x) = 35x^6$

Question 5

Let $$f(x) = x^3$ and g(x$$) = \sin x$$. Using the product rule, what is the derivative of h(x$$) = $$x^3$$ \sin x$$?

Accepted Answer

$$h'(x) = 3x^2$ \sin x + x^3$ \cos x$$

Question 6

A pharmaceutical company models the concentration of a drug in the bloodstream as $$C(t) = 2e$$^{0.5t}$$. What is the rate of change of the concentration at any time t$?

Accepted Answer

$$C'(t) = 2 \cdot 0.5 e$$^{0.5t}$$ = e$$^{0.5t}$$

Question 7

If $$h(x) = \frac{3x^2$}{\sqrt{x} e$$^{x+1}$$}$$, what is the first step in applying the quotient rule to find $$h'(x)$$?

Accepted Answer

Let $$f(x) = 3x^2$, g(x$$) = \sqrt{x} $$e^{x+1}$$, then apply $$h'(x) = \frac{g(x)f'(x) - f(x)g'(x)}{[g(x)]^2}$$

Question 8

Which of the following correctly expresses the second derivative of $$f(x) = x^5$ - x^3$ + 1$$?

Accepted Answer

$$f''(x) = 20x^3$ - 6x$$

Question 9

A ball is thrown vertically and its height at time $$t is $$given by $$s(t) = -4.9t^2$ + 20t + 1. $$What is the acceleration of the ball at any time $$t$$?

Accepted Answer

$$a(t) = -9.8$$

Question 10

Which of the following is the correct derivative of $$f(x) = \sin x$$?

Accepted Answer

$$f'(x) = \cos x$$

Question 11

If $$y = 	an x$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{dy}{dx} = \sec^2$ x$$

Question 12

Find the third derivative of $$f(x) = x^5$ - x^3$ + 1.$$

Accepted Answer

$$f^{(3)}(x$$) = $$60x^2 - 6$

Question 13

Which of the following statements about higher derivatives is TRUE?

Accepted Answer

$$f^{(0)}(x$$)$$ is the original function f(x$$)$$.

Question 14

If $$f(x) = \cos x$$, what is $$f^{(4)}(x$$)$$?

Accepted Answer

$$f^{(4)}(x$$) = \cos x$$

Question 15

A population of bacteria grows according to $$P(t) = 100e$$^{0.03t}$$. What is the instantaneous rate of growth at t = 0$?

Accepted Answer

$$P'(0) = 3$$

Question 16

Which of the following is the correct derivative of $$f(x) = \sec x$$?

Accepted Answer

$$f'(x) = \sec x 	an x$$

Question 17

If $$f(x) = x$$^{\pi}$$, what is f$$'(x)$$?

Accepted Answer

$$f'(x) = \pi x$$^{\pi-1}$$

Question 18

Suppose $$f(x) = 3x^5$ - 4x^2$ + x + 1. $$What is $$f'(x)$$?

Accepted Answer

$$f'(x) = 15x^4$ - 8x + 1$$

Derivative Rules and Higher Derivatives

Study Guide - Practice Questions

Study Guide - Flashcards