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

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

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