Question 1

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

Accepted Answer

$$f'(x) = 0$$

Question 2

A medication's concentration in the bloodstream is modeled by $$C(t) = 5e$$^{0.2t}$$. What is the rate of change of the concentration at time t$?

Accepted Answer

$$C'(t) = 5 \cdot 0.2 e$$^{0.2t}$$

Question 3

Given $$f(x) = x^4$ - 3x^2$ + 2$$, what is $$f'(x)$$?

Accepted Answer

$$f'(x) = 4x^3$ - 6x$$

Question 4

If $$y = x^2$ e^x$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$x^2 e^x + 2x e^x$

Question 5

A population of bacteria grows according to $$P(t) = 100e$$^{0.05t}$$. What is the instantaneous rate of growth at t = 10$?

Accepted Answer

$$P'(10) = 0.05 \cdot 100e$$^{0.05 \cdot 10}$$

Question 6

Which of the following is the correct application of the product rule for $$f(x) = x^3$ \sin x$$?

Accepted Answer

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

Question 7

Find the derivative of $$y = \frac{x^2$ - 4x + 2}{x - 1}.$$

Accepted Answer

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

Question 8

If $$f(x) = \sqrt{x}g(x)$$ and $$g(4) = 2$$, $$g'(4) = 3$$, what is $$f'(4)$$?

Accepted Answer

$$f'(4) = 2 \cdot 3 + \frac{1}{4} \cdot 2$$

Question 9

A ball is thrown upward and its height is given by $$h(t) = 10t - 4.9t^2$. What is the velocity of the ball at t = 2$ seconds?

Accepted Answer

$$h'(2) = 10 - 9.8$$

Question 10

Which of the following is the correct derivative of $$y = \cos x$$?

Accepted Answer

$$y' = -\sin x$$

Question 11

Find the equation of the tangent line to $$y = x + \frac{2}{x} at $$the point $$(2, 3).$$

Accepted Answer

$$y = \frac{1}{2}(x-2) + 3$$

Question 12

If $$f(x) = 	an x$$, what is $$f'(x)$$?

Accepted Answer

$$f'(x) = \sec^2$ x$$

Question 13

A chemical reaction's rate is modeled by $$R(t) = 3t^2$ - 2t + 5. $$What is the instantaneous rate of change at $$t = 4$$?

Accepted Answer

$$R'(4) = 6 \cdot 4 - 2$$

Question 14

Which of the following is the correct derivative of $$f(x) = \frac{1}{x}$$?

Accepted Answer

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

Question 15

If $$y = e^x + x^6$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$e^x + 6x^5$

Question 16

Which of the following is the correct application of the quotient rule for $$y = \frac{\cos x}{1 + 2\sin x}$$?

Accepted Answer

$$y' = \frac{(1+2\sin x)(-\sin x) - (\cos x)(2\cos x)}{(1+2\sin x)^2$}$$

Question 17

If $$f(x) = x^2$ + 3x$$ and $$g(x) = x^3$, what is $$\frac{d}{dx}[f(x)g(x)]$$?

Accepted Answer

$$x^2$$ \cdot $$3x^2$$ + $$(2x+3)x^3$$

Question 18

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

Accepted Answer

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

Question 19

A company's profit is modeled by $$P(x) = 2x^3$ - 5x^2$ + 4x - 7. $$What is the marginal profit when $$x = 2$$?

Accepted Answer

$$P'(2) = 6 \cdot 2^2$ - 10 \cdot 2 + 4$$

Differentiation Rules and Applications in Calculus

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