Question 1

Which of the following is the correct derivative of $$f(x) = 6^x$$?

Accepted Answer

$$6^x \cdot \ln 6$$

Question 2

A medical lab uses a radioactive isotope whose quantity after $$t$$ hours is $$R = 150 \left(\frac{1}{3}\$$right)^{t/12}$$. What is the instantaneous rate of change $$\frac{dR}{dt}$$ at t=12$ hours?

Accepted Answer

$$150 \left(\frac{1}{3}\right) \cdot \frac{1}{12} \cdot \ln\left(\frac{1}{3}\right)$$ evaluated at $$t=12$$ gives $$\approx -13.75$$

Question 3

Find the derivative of $$f(x) = x^4$ + 4x + 4^x.$$

Accepted Answer

$$4x^3 + 4 + 4^x$$ \cdot \ln 4$$

Question 4

Which rule should be used to find the derivative of $$y = (x^2$ + 2x) \cdot \log_5$ x$$?

Accepted Answer

Product rule and general logarithmic rule

Question 5

Given $$f(x) = e$$^{x^2 + 4x}$$, what is f$$'(x)$$?

Accepted Answer

$$e^{x^2 + 4x}$$ \cdot (2x + 4)$$

Question 6

Find the derivative of $$g(x) = \log_2$(7^{2x+4}$$)$.

Accepted Answer

$$\frac{2 \ln 7}{\ln 2}$$

Question 7

A company models its revenue by $$R(x) = 2x^2$ - 1 + \ln x. $$What is the marginal revenue at $$x=1$$?

Accepted Answer

$$4x - \frac{1}{x}$$ evaluated at $$x=1$$ gives $$4 - 1 = 3$$

Question 8

Which of the following is the correct derivative of $$y = e$$^{\sqrt{x^2 - x}$$}$$?

Accepted Answer

$$e^{\sqrt{x^2 - x}$$} \cdot \frac{2x - 1}{2\sqrt{$$x^2 - x$$}}$$

Question 9

Find the equation of the tangent line to $$y = 3e^x - x at$$ $$x = 0.$$

Accepted Answer

$$y = 2x + 2$$

Question 10

Which of the following is the correct derivative of $$h(x) = \ln\left(\frac{\sqrt{x+1}}{x^2$+3}\right)$$?

Accepted Answer

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

Question 11

For the function $$y = x$$^{x}$$, which differentiation technique is necessary?

Accepted Answer

Logarithmic differentiation

Question 12

Find the derivative of $$f(x) = x^2$ e$$^{1+x}$$.

Accepted Answer

$$2x e$$^{1+x}$$ + x^2$ e$$^{1+x}$$

Question 13

Given $$g(x) = x e$$^{x \ln x}$$, what is g$$'(x)$$?

Accepted Answer

$$e^{x \ln x} + x$$ $$e^{x \ln x}$$ (\ln x + 1)$$

Question 14

A biologist models the time $$t(B) = 42 \ln\left(\frac{650}{1500-B}\right) to $$reach $$B$$ bacteria. What does $$t'(1000)$$ represent?

Accepted Answer

The rate at which time changes as the number of bacteria increases at $$B=1000$$

Question 15

Find the derivative of $$f(x) = 8 \log_2$ x.$$

Accepted Answer

$$\frac{8}{x \ln 2}$$

Question 16

Which of the following is the correct derivative of $$y = \ln(x^2$ + 4x)$$?

Accepted Answer

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

Question 17

Use logarithmic differentiation to find the derivative of $$y = \sqrt{\frac{(x+2)^3$}{x(x^2$-4)}}.$$

Accepted Answer

$$\frac{1}{2} \left( \frac{3}{x+2} - \frac{1}{x} - \frac{2x}{x^2$-4} \right) \sqrt{\frac{(x+2)^3$}{x(x^2$-4)}}$$

Derivatives of Exponential and Logarithmic Functions in Business Calculus

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