$y' = \sec(x e^x) \tan(x e^x) \cdot (e^x + x e^x)$

Question 1

Which of the following is the correct application of the chain rule to find the derivative of $y = \sin(12x - 7)$?

Accepted Answer

$y' = 12 \cos(12x - 7)$

Question 2

A biologist models the population of bacteria as $y = e^{3x^2 + 1}$. What is $\frac{dy}{dx}$?

Accepted Answer

$y' = 6x e^{3x^2 + 1}$

Question 3

Given $y = 	an(e^x)$, which of the following is the correct derivative?

Accepted Answer

$y' = \sec^2(e^x) \cdot e^x$

Question 4

A physicist models the displacement of a spring as $y = (3x^2 + 2)^6$. What is $\frac{dy}{dx}$?

Accepted Answer

$y' = 36x(3x^2 + 2)^5$

Question 5

Find the derivative of $y = \frac{2}{(9x + 3)^3}$ using the chain rule.

Accepted Answer

$y' = -54(9x + 3)^{-4}$

Question 6

A chemical reaction rate is modeled by $y = \left(\frac{6x}{5x - 2}\right)^4$. What is $\frac{dy}{dx}$?

Accepted Answer

$y' = 4 \left(\frac{6x}{5x - 2}\right)^3 \cdot \frac{30x - 12 - 36x}{(5x - 2)^2}$

Question 7

Find the derivative of $y = \sqrt{8x - 3}$ using the chain rule.

Accepted Answer

$y' = 4(8x - 3)^{-1/2}$

Question 8

A medical researcher models the concentration of a drug as $y = \sec(x e^x)$. What is $\frac{dy}{dx}$?

Accepted Answer

$y' = \sec(x e^x) 	an(x e^x) \cdot (e^x + x e^x)$

Question 9

Find the derivative of $y = 	an(\sin(8x + 1))$ using the chain rule.

Accepted Answer

$y' = \sec^2(\sin(8x + 1)) \cdot \cos(8x + 1) \cdot 8$

Question 10

Given $y = \sin^3 x + \sin x^3$, what is $\frac{dy}{dx}$?

Accepted Answer

$y' = 3\sin^2 x \cos x + \cos x^3 \cdot 3x^2$

Question 11

A population model is given by $y = (5x - 1)^2 e^{4x}$. What is $\frac{dy}{dx}$?

Accepted Answer

$y' = 10(5x - 1) e^{4x} + 4(5x - 1)^2 e^{4x}$

Question 12

Find the derivative of $y = \frac{\sin 3x}{2x - 1}$ using the quotient rule and chain rule.

Accepted Answer

$y' = \frac{3(2x - 1) \cos 3x - 2 \sin 3x}{(2x - 1)^2}$

Question 13

Which of the following best describes the chain rule for differentiating $f(u(x))$?

Accepted Answer

$f'(u(x)) \cdot u'(x)$

Question 14

A pharmaceutical company models the concentration of a drug as $y = 	an(e^{x^2})$. What is $\frac{dy}{dx}$?

Accepted Answer

$y' = \sec^2(e^{x^2}) \cdot 2x e^{x^2}$

Question 15

Find the derivative of $y = \cos(5x^2 - 3x)$ using the chain rule.

Accepted Answer

$y' = -\sin(5x^2 - 3x) \cdot (10x - 3)$

Question 16

A physicist models the velocity of a particle as $y = e^{\sin x}$. What is $\frac{dy}{dx}$?

Accepted Answer

$y' = e^{\sin x} \cos x$

Question 17

Find the derivative of $y = (2x^3 + 1)^5$ using the chain rule.

Accepted Answer

$y' = 5(2x^3 + 1)^4 \cdot 6x^2$

Question 18

A biologist models the growth of a cell as $y = \sin(e^{2x})$. What is $\frac{dy}{dx}$?

Accepted Answer

$y' = \cos(e^{2x}) \cdot 2e^{2x}$

The Chain Rule and Differentiation of Composite Functions: 3.7

Study Guide - Practice Questions