Question 1

Which of the following is the correct derivative of $y = \log_b g(x)$ with respect to $x$?

Accepted Answer

$y' = \dfrac{g'(x)}{g(x) \ln b}$

Question 2

Given $y = \ln(2x^4 + x)$, what is $\dfrac{dy}{dx}$?

Accepted Answer

$\dfrac{8x^3 + 1}{2x^4 + x}$

Question 3

If $f(x) = \log_5(4 + \cos x^3)$, what is $f'(x)$?

Accepted Answer

$\dfrac{-2x \sin x^3}{(4 + \cos x^3) \ln 5}$

Question 4

Find $\dfrac{d}{dx} \left( \ln \dfrac{x+1}{\sqrt{x-2}} \right)$.

Accepted Answer

$\dfrac{x+1}{x-2} - \dfrac{1}{2} \cdot \dfrac{x+1}{(x-2)^{3/2}}$

Question 5

Which of the following is the correct derivative of $y = \sin^{-1}(x)$?

Accepted Answer

$\dfrac{1}{\sqrt{1-x^2}}$

Question 6

Given $y = 2(\sin^{-1} x)^2$, find $\dfrac{dy}{dx}$ using the chain rule.

Accepted Answer

$\dfrac{4 \sin^{-1} x}{\sqrt{1-x^2}}$

Question 7

Find $\dfrac{d}{dx} \left( \cos^{-1}(x^3 + 2) \right)$.

Accepted Answer

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

Question 8

Which formula gives the derivative of $y = 	an^{-1}(g(x))$?

Accepted Answer

$\dfrac{g'(x)}{1 + (g(x))^2}$

Question 9

Suppose $y = \sec^{-1}(f(x))$. What is $\dfrac{dy}{dx}$?

Accepted Answer

$\dfrac{f'(x)}{f(x) \sqrt{f(x)^2 - 1}}$

Question 10

For the equation $x^2 + y^2 = 4$, which method should be used to find $\dfrac{dy}{dx}$?

Accepted Answer

Implicit differentiation

Question 11

Using implicit differentiation, find $\dfrac{dy}{dx}$ for $x^2 + y^2 = 4$.

Accepted Answer

$\dfrac{-x}{y}$

Question 12

Given $x^2 + 4xy^2 + y^2 = 4$, use implicit differentiation to find $\dfrac{dy}{dx}$.

Accepted Answer

$\dfrac{-2x - 4y}{4x y + 2y}$

Question 13

For the folium of Descartes $x^3 + y^3 = 6xy$, what is $\dfrac{dy}{dx}$?

Accepted Answer

$\dfrac{3x^2 - 6y}{6x - 3y^2}$

Question 14

At the point $(3,3)$ on the folium $x^3 + y^3 = 6xy$, what is the slope of the tangent line?

Accepted Answer

$0$

Question 15

For which value of $y$ is the tangent to the folium $x^3 + y^3 = 6xy$ horizontal in the first quadrant?

Accepted Answer

$y = \dfrac{1}{2}x^2$

Question 16

Which of the following is the correct derivative of $y = \cot^{-1}(g(x))$?

Accepted Answer

$\dfrac{-g'(x)}{1 + (g(x))^2}$

Question 17

If $y = \csc(x^2)$, what is $\dfrac{dy}{dx}$?

Accepted Answer

$-\csc(x^2) \cot(x^2) \cdot 2x$

Question 18

Which rule is most appropriate for differentiating $y = \ln(g(x))$?

Accepted Answer

Chain rule

Derivatives of Logarithmic, Inverse Trigonometric Functions, and Implicit Differentiation

Study Guide - Practice Questions