Question 1

Which of the following is the correct derivative of the exponential function $f(x) = a^x$, where $a > 0$ 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 $g(x) > 0$?

Accepted Answer

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

Question 5

Let $f(x) = \log_a(x)$, where $a > 0$ 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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