$$21 (\tan x + 10$$)^{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

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