Question 1

Which of the following is the correct derivative of the exponential function $$f(x) = a^x$$, where $$a \u003e 0$$ and $$a 
eq 1$$?

Accepted Answer

$$f'(x) = a^x \ln a$$

Question 2

Suppose the population of a species is modeled by the logistic function $$P(t) = \frac{m P_0$}{P_0$ + (m - P_0$) e$$^{-kmt}$$}. $$What is the limiting value of $$P(t) as$$ $$t 	o \infty$$?

Accepted Answer

$$m$$

Question 3

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

Accepted Answer

$$f'(x) = 2x e$$^{x^2}$$

Question 4

Which of the following is the correct formula for the derivative of $$f(x) = \ln(g(x))$$, where $$g(x) \u003e 0$$?

Accepted Answer

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

Question 5

Let $$f(x) = 3x^2$ \cdot \log_3$ x. $$Find $$f'(x).$$

Accepted Answer

$$f'(x) = 6x \cdot \log_3$ x + 3x^2$ \cdot \frac{1}{x \ln 3}$$

Question 6

If $$f(x) = \ln|x|$$, what is $$f'(x)$$ for $$x 
eq 0$$?

Accepted Answer

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

Question 7

Find the derivative of $$f(x) = a$$^{g(x)}$$, where a$$ \u003e 0$$ and g(x$$)$$ is differentiable.

Accepted Answer

$$f'(x) = (\ln a) a$$^{g(x)}$$ g'(x)$$

Question 8

Suppose $$y = \ln(kx)$$, where $$k is a $$constant and $$k 
eq 0. $$What is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{1}{x}$$

Question 9

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

Accepted Answer

$$f'(x) = \frac{1}{x \ln a}$$

Question 10

Let $$f(x) = x e^x. $$Find $$f'(x).$$

Accepted Answer

$$f'(x) = e^x + x e^x$$

Question 11

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

Accepted Answer

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

Question 12

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

Accepted Answer

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

Question 13

Let $$f(x) = \frac{3}{10 + e$$^{-2x}$$}. $$Find $$f'(x).$$

Accepted Answer

$$f'(x) = \frac{6 e$$^{-2x}$$}{(10 + e$$^{-2x}$$)^2$$}$

Question 14

Suppose $$f(x) = \ln(\sqrt{x^2$ + x}). $$What is $$f'(x)$$?

Accepted Answer

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

Question 15

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

Accepted Answer

$$f'(x) = \frac{1}{2(2 + 3x^4$)^{1/2}$$} \cdot $$(12x^3$$)$

Question 16

If $$f(x) = \log_5$(x^3 + 1$$)$$, what is f$$'(x)$?

Accepted Answer

$$f'(x) = \frac{3x^2$}{(x^3$ + 1) \ln 5}$$

Question 17

Let $$f(x) = e$$^{x}$$ \cdot x. $$Which rule is most appropriate to find $$f'(x)$$?

Accepted Answer

Product rule

Question 18

Suppose $$g(x) is $$the inverse of $$f(x)$$ and $$f'(g(x)) = 2x. $$What is $$g'(x)$$?

Accepted Answer

$$g'(x) = \frac{1}{2x}$$

Derivatives of Exponential and Logarithmic Functions in Business Calculus

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