Question 1

Which of the following is the correct derivative of the function $$f(x) = x^3$ \sin(x)$$?

Accepted Answer

$$3x^2$$ \sin(x) + $$x^3$$ \cos(x)$$

Question 2

A biologist models the population of a bacteria culture by $$P(t) = 500e$$^{0.04t}$$, where t$ is in hours. What is the instantaneous rate of change of the population at t = 10$ hours?

Accepted Answer

$$P'(10) = 20e$$^{0.4}$$ \approx 29.84$$

Question 3

Find the equation of the tangent line to the curve $$y = \ln(x^2$ + 1) at $$the point where $$x = 1.$$

Accepted Answer

Question 4

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

Accepted Answer

$$2x \cos(x^2$)$$

Question 5

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

Accepted Answer

$$\cot(x)$$

Question 6

A medication is administered so that its concentration in the bloodstream is $$C(t) = 10e$$^{-0.5t}$$ mg/L, where t$ is in hours. What is the rate at which the concentration is changing at t = 2$ hours?

Accepted Answer

$$C'(2) = -5e$$^{-1}$$

Question 7

Find the second derivative of $$f(x) = e$$^{2x}$$.

Accepted Answer

$$4e^{2x}$$

Question 8

Which of the following is the correct derivative of $$f(x) = \log_b(x)$$, where $$b \u003e 0$$ and $$b 
eq 1$$?

Accepted Answer

$$\frac{1}{x \ln(b)}$$

Question 9

If $$y = 	an$$^{-1}$$(x)$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{1}{1 + x^2$}$$

Question 10

A car's position is given by $$s(t) = t^2$ \ln(t)$$, where $$t \u003e 0 is $$time in seconds. What is the velocity of the car at $$t = 1$$ second?

Accepted Answer

$$v(1) = 1$$

Question 11

Which of the following is the correct application of the chain rule for $$\frac{d}{dx} \left[ \sin(3x^2$) \right]$$?

Accepted Answer

$$6x \cos(3x^2$)$$

Question 12

If $$f(x) = \ln(x^2$ + 4)$$, what is $$f'(x)$$?

Accepted Answer

$$\frac{2x}{x^2$ + 4}$$

Question 13

Which of the following is the correct derivative of $$f(x) = e$$^{x^2}$$?

Accepted Answer

$$2x e$$^{x^2}$$

Question 14

A chemical reaction's rate is modeled by $$R(t) = 5 \ln(2t + 1)$$, where $$t is in $$minutes. What is the instantaneous rate of change of $$R at$$ $$t = 2$$?

Accepted Answer

$$R'(2) = 2$$

Question 15

Find the derivative of $$f(x) = x^x$$ for $$x \u003e 0.$$

Accepted Answer

$$x^x (\ln(x) + 1)$$

Question 16

Which of the following is the correct derivative of $$f(x) = \cos(x) \cdot e^x$$?

Accepted Answer

$$-\sin(x) e^x + \cos(x) e^x$$

Question 17

If $$y = \ln(\ln(x))$$, what is $$\frac{dy}{dx}$$ for $$x \u003e 1$$?

Accepted Answer

$$\frac{1}{x \ln(x)}$$

Question 18

A company's revenue is given by $$R(x) = 1000 \cdot \log_{10}(x)$$, where $$x is $$the number of units sold. What is the marginal revenue when $$x = 100$$?

Accepted Answer

$$R'(100) = 4.3429$$

Question 19

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

Accepted Answer

$$-\frac{1}{x^2$}$$

Calculus I: Derivatives – Rules, Trigonometric, Exponential, and Logarithmic Functions

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