Question 1

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

Accepted Answer

$$\frac{d}{dx} \ln(x) = \frac{1}{x}$$

Question 2

Given $$f(x) = e$$^{5x}$$, what is f$$'(x)$$?

Accepted Answer

$$f'(x) = 5e$$^{5x}$$

Question 3

If $$f(x) = x^5$ e$$^{5x}$$, which rule is most appropriate to use to differentiate f(x$$)$$?

Accepted Answer

Product Rule

Question 4

Differentiate $$f(x) = x^5$ e$$^{5x}$$.

Accepted Answer

$$f'(x) = x^5$ e$$^{5x}$$ (5) + 5x^4$ e$$^{5x}$$

Question 5

Which of the following statements is true about $$e^x$$ and $$\ln(x)$$?

Accepted Answer

$$e^x$$ and $$\ln(x)$$ are inverse functions

Question 6

Find the derivative of $$f(x) = \ln(3x^2$).$$

Accepted Answer

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

Question 7

Suppose $$f(x) = \frac{e$$^{5x}$$}{x^3$}. $$What is $$f'(x)$$?

Accepted Answer

$$f'(x) = \frac{5x^3$ e$$^{5x}$$ - 3x^2$ e$$^{5x}$$}{x^6$}$$

Question 8

Which rule is used to differentiate $$f(x) = \frac{e$$^{5x}$$}{x^3$}$$?

Accepted Answer

Quotient Rule

Question 9

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

Accepted Answer

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

Question 10

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

Accepted Answer

$$C'(2) = -0.5e$$^{-1}$$

Question 11

Which of the following best describes the chain rule for differentiating composite functions?

Accepted Answer

Differentiate the outer function, then multiply by the derivative of the inner function

Question 12

If $$f(x) = e$$^{g(x)}$$, what is f$$'(x)$$?

Accepted Answer

$$f'(x) = e$$^{g(x)}$$ \cdot g'(x)$$

Question 13

A company’s revenue is modeled by $$R(x) = 100 \ln(x)$$, where $$x is $$the number of units sold. What is the marginal revenue when $$x = 10$$?

Accepted Answer

$$R'(10) = 10$$

Question 14

Which of the following is the correct application of the product rule to $$f(x) = u(x)v(x)$$?

Accepted Answer

$$f'(x) = u'(x)v(x) + u(x)v'(x)$$

Derivatives of Exponential and Logarithmic Functions

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