Question 1

Which of the following is the correct derivative of the function $$f(x) = 7x^5$ - 3x^2$ + 4$$?

Accepted Answer

$$f'(x) = 35x^4$ - 6x$$

Question 2

A medication's concentration in the bloodstream is modeled by $$C(t) = 5e^t - 2t^2$. What is the rate of change of concentration at time t$?

Accepted Answer

$$C'(t) = 5e^t - 4t$$

Question 3

Which rule must be used to differentiate $$y = (3x^2$ + 2)(x^3$ - 1)$$?

Accepted Answer

Product Rule

Question 4

Find the derivative of $$f(x) = \frac{4x^2$ + 1}{x - 3}.$$

Accepted Answer

$$f'(x) = \frac{(x-3)(8x) - (4x^2$+1)(1)}{(x-3)^2$}$$

Question 5

If $$y = 2e^x + 5x^2$, what is the equation of the tangent line at x = 0$?

Accepted Answer

$$y = 2x + 2$$

Question 6

Which of the following statements about the derivative of a constant function is correct?

Accepted Answer

The derivative is always zero.

Question 7

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

Accepted Answer

Question 8

Which of the following is the correct application of the product rule for $$y = x^2$ e^x$$?

Accepted Answer

$$y' = x^2$ e^x + 2x e^x$$

Question 9

Find the $$x-$$values where the tangent line to $$y = x^3$ - 3x - 2 is $$horizontal.

Accepted Answer

$$x = 1, -1$$

Question 10

Given $$f(x) = 6e^x - 2x^2$ + 3$$, what is $$f''(x)$$?

Accepted Answer

$$f''(x) = 6e^x - 4$$

Question 11

A company's profit is modeled by $$P(x) = x^2$ e^x. $$What is the second derivative $$P''(x)$$?

Accepted Answer

$$P''(x) = x^2$ e^x + 4x e^x + 2e^x$$

Question 12

Which of the following is a common misconception about the derivative of a product?

Accepted Answer

$$\frac{d}{dx}[f(x)g(x)] = f'(x)g'(x)$$

Question 13

Find the derivative of $$y = \frac{6e^x}{1 + e^x}.$$

Accepted Answer

$$y' = \frac{6e^x}{(1 + e^x)^2$}$$

Question 14

Which of the following best describes the sum rule for derivatives?

Accepted Answer

$$\frac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x)$$

Question 15

A chemical reaction's rate is modeled by $$R(t) = 4t^3$ - 2t + 7. $$What is the instantaneous rate of change at $$t = 2$$?

Accepted Answer

$$R'(2) = 4 \cdot 3 \cdot 2^2$ - 2 = 48 - 2 = 46$$

Question 16

Which of the following is the correct application of the quotient rule for $$f(x) = \frac{x^2$ e^x}{x^3$ + 5e^x}$$?

Accepted Answer

$$f'(x) = \frac{(x^3$ + 5e^x)(2x e^x + x^2$ e^x) - x^2$ e^x (3x^2$ + 5e^x)}{(x^3$ + 5e^x)^2$}$$

Differentiation Rules and Applications in Calculus

Study Guide - Practice Questions