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

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

Question 2

Suppose a population grows according to 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 $$f(x) = \log_a(x)$$, where $$a \u003e 0$$ and $$a 
eq 1. $$What is $$f'(x)$$?

Accepted Answer

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

Question 6

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

Accepted Answer

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

Question 7

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 8

Which of the following is the correct derivative of $$f(x) = \ln|x|$$ for $$x 
eq 0$$?

Accepted Answer

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

Question 9

Let $$f(x) = x e^x. $$What is $$f'(x)$$?

Accepted Answer

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

Question 10

Suppose $$f(x) = 3x^2$ + e$$^{-2x}$$. What is f$$'(x)$$?

Accepted Answer

$$f'(x) = 6x - 2e$$^{-2x}$$

Question 11

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

Accepted Answer

$$f'(x) = 2x \ln x + x$$

Question 12

If $$y = \ln(kx)$$ where $$k 
eq 0 is a $$constant, what is $$\frac{dy}{dx}$$?

Accepted Answer

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

Question 13

Let $$f(x) = x^2$ e$$^{-x}$$. What is f$$'(x)$$?

Accepted Answer

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

Question 14

Which of the following is the correct derivative of $$f(x) = \log_a|g(x)|$$, where $$g(x) 
eq 0$$?

Accepted Answer

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

Question 15

Suppose $$f(x) = 5$$^{\sqrt{t}$$}. $$What is $$\frac{d}{dt} f(t)$$?

Accepted Answer

$$\frac{d}{dt} f(t) = (\ln 5) \cdot 5$$^{\sqrt{t}$$} \cdot \frac{1}{2\sqrt{t}}$$

Question 16

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

Accepted Answer

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

Question 17

Let $$f(x) = 3x^3$ \cdot \log_3$ x. $$What is $$f'(x)$$?

Accepted Answer

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

Question 18

If $$g(x) is $$the inverse of $$f(x)$$ and $$f'(g(x)) is $$known, what is $$g'(x)$$?

Accepted Answer

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

Question 19

Which of the following statements about the derivative of $$f(x) = e^x is $$correct?

Accepted Answer

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

(L12) Derivatives of Exponential and Logarithmic Functions in Business Calculus

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