Question 1

Given $f(x) = \frac{\cos(x)}{x^2}$, what is $f'(x)$?

Accepted Answer

$f'(x) = -\frac{\sin(x)}{x^2} - \frac{2\cos(x)}{x^3}$

Question 2

If $y = 3x^{5/2} + 3x^{1/5} + \ln(x^5 + 1) + 5x^3 + x^n$, what is $\frac{dy}{dx}$?

Accepted Answer

$\frac{15}{2}x^{3/2} + \frac{3}{5}x^{-4/5} + \frac{5x^4}{x^5 + 1} + 15x^2 + nx^{n-1}$

Question 3

Which of the following is the correct application of the quotient rule for $f(x) = \frac{g(x)}{h(x)}$?

Accepted Answer

$f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2}$

Question 4

Given $y = \ln(x^2 + 1)$, what is $\frac{dy}{dx}$?

Accepted Answer

$\frac{2x}{x^2 + 1}$

Question 5

If $y = \ln(x) + x^2$, what is $\frac{dy}{dx}$?

Accepted Answer

$\frac{1}{x} + 2x$

Question 6

A medication's concentration in the bloodstream is modeled by $C(t) = \frac{e^{-kt}}{t}$, where $k > 0$. What is $C'(t)$?

Accepted Answer

$C'(t) = -\frac{ke^{-kt}}{t} - \frac{e^{-kt}}{t^2}$

Question 7

Suppose $f(x) = x^2 \sin(x)$. What is $f'(x)$?

Accepted Answer

$f'(x) = 2x\sin(x) + x^2\cos(x)$

Question 8

Given $y = \ln(x^2 + 1) + 3x^5$, what is $\frac{dy}{dx}$ evaluated at $x = 1$?

Accepted Answer

$\frac{2}{2} + 3 \cdot 5 \cdot 1^4 = 1 + 15 = 16$

Question 9

If $y = \ln(x^2 + 1)$, what is the slope of the tangent line at $x = 0$?

Accepted Answer

$0$

Question 10

A population grows according to $P(t) = 5e^{0.03t}$. What is the instantaneous rate of change at $t = 10$?

Accepted Answer

$P'(10) = 0.15e^{0.3}$

Question 11

Given $f(x) = \frac{2x\cos^2(x) + (x^2 + 1)\sin(x)\cos(x)}{\cos^2(x)}$, which of the following simplifies $f'(x)$?

Accepted Answer

$f'(x) = 2x\cos(x) + (x^2 + 1)\sin(x)$

Question 12

If $y = \ln(x^2 + 1)$, what is the second derivative $\frac{d^2y}{dx^2}$?

Accepted Answer

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

Question 13

A chemical reaction's rate is given by $R(t) = \ln(t^2 + 1)$. What is the rate of change of $R$ at $t = 2$?

Accepted Answer

$\frac{4}{5}$

Question 14

If $y = \ln(x^2 + 1)$, for which $x$ is $\frac{dy}{dx} = 1$?

Accepted Answer

$x = 1$

Question 15

Given $y = x^n$, which of the following is the correct derivative?

Accepted Answer

$y' = nx^{n-1}$

Question 16

Suppose $y = \ln(x^2 + 1) + 3x^5$. What is the value of $\frac{dy}{dx}$ at $x = 0$?

Accepted Answer

$0$

Question 17

If $y = \ln(x^2 + 1)$, what is the value of $\frac{dy}{dx}$ as $x 	o \infty$?

Accepted Answer

$0$

Question 18

Given $y = \ln(x^2 + 1)$, which of the following best describes the concavity of $y$ for $x > 0$?

Accepted Answer

Concave down for $x > 1$, concave up for $0 < x < 1$

Calculus Study Notes: Derivatives of Quotients and Logarithmic Functions

Study Guide - Practice Questions