$21 (\tan x + 10)^{20} \sec^2 x$

Question 1

Which of the following best describes the Chain Rule for differentiating composite functions?

Accepted Answer

$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$

Question 2

Given $y = \sin(e^{\cos x})$, what is $\frac{dy}{dx}$?

Accepted Answer

$-\sin x \cdot e^{\cos x} \cdot \cos(e^{\cos x})$

Question 3

For $y = (5x + 4)^{1/2}$, what is the derivative $\frac{dy}{dx}$?

Accepted Answer

$\frac{5}{2} \sqrt{5x + 4}$

Question 4

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

Accepted Answer

$2x \cos(x^2)$

Question 5

Which step is necessary when using the Chain Rule to differentiate $f(g(x))$?

Accepted Answer

Multiply the derivative of the outer function evaluated at the inner function by the derivative of the inner function.

Question 6

Find $\frac{d}{dx} \left( x^2 \sqrt{x^2 + 1} \right)$.

Accepted Answer

$\frac{3x^2 + 2x}{\sqrt{x^2 + 1}}$

Question 7

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

Accepted Answer

$3 \sin^2 x \cos x$

Question 8

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

Accepted Answer

$3x^2 \cos(x^3)$

Question 9

Which of the following is the correct application of the Chain Rule for powers?

Accepted Answer

$\frac{d}{dx} \left( (g(x))^p \right) = p (g(x))^{p-1} g'(x)$

Question 10

Find $\frac{d}{dx} \left( (	an x + 10)^{21} ight)$.

Accepted Answer

$21 (	an x + 10)^{20} \sec^2 x$

Question 11

A medical researcher models the growth of bacteria as $N(t) = e^{2t^3}$. What is the rate of change of bacteria population with respect to time?

Accepted Answer

$6t^2 e^{2t^3}$

Question 12

If $y = \sqrt{3x^2 + 2x}$, what is $\frac{dy}{dx}$?

Accepted Answer

$\frac{6x + 2}{2 \sqrt{3x^2 + 2x}}$

Question 13

A chemical reaction rate is modeled by $R(t) = \ln(5t^2 + 1)$. What is $\frac{dR}{dt}$?

Accepted Answer

$\frac{10t}{5t^2 + 1}$

Question 14

If $y = e^{\sin x}$, what is $\frac{dy}{dx}$?

Accepted Answer

$e^{\sin x} \cos x$

Question 15

Which of the following is NOT a correct step in the Chain Rule procedure?

Accepted Answer

Add the derivatives of the inner and outer functions.

Question 16

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

Accepted Answer

$10 (2x + 3)^4$

Question 17

A population model is $P(t) = \cos(4t^2)$. What is $\frac{dP}{dt}$?

Accepted Answer

$-8t \sin(4t^2)$

Question 18

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

Accepted Answer

$\frac{6x^2}{2x^3 + 1}$

Question 19

Which real-world scenario best illustrates the Chain Rule?

Accepted Answer

Calculating how fast computer A is compared to computer C, given A is 10 times faster than B and B is 15 times faster than C.

The Chain Rule: Derivatives of Composite Functions

Study Guide - Practice Questions