Q: Which of the following best describes the Chain Rule for differentiation?

$$\text{If } F(x) = f(g(x)), \text{ then } F'(x) = f'(g(x)) \cdot g'(x)$$

Q: Find the derivative of $$y = e$$^{\sec 3\theta}$$ with respect to $$\theta$$.

$$3e^{\sec 3\theta}$$ \sec 3\theta \tan 3\theta$$

Question 1

Which of the following best describes the Chain Rule for differentiation?

Accepted Answer

$$	ext{If } F(x) = f(g(x)), 	ext{ then } F'(x) = f'(g(x)) \cdot g'(x)$$

Question 2

Find the derivative of $$y = \sin(x^2$).$$

Accepted Answer

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

Question 3

Differentiate $$y = (x^3$ - 1$$)^{100}$$.

Accepted Answer

$$300x^2(x^3$ - 1$$)^{99}$

Question 4

If $$f(x) = \frac{1}{\sqrt[3]{x^2$ + x + 1}}$$, what is $$f'(x)$$?

Accepted Answer

$$-\frac{1}{3}(x^2$ + x + 1$$)^{-4/3}$$(2x + 1)$$

Question 5

Find the derivative of $$g(t) = \left(\frac{t-2}{2t+1}\$$right)^9$$.

Accepted Answer

$$\frac{45(t-2)^8$}{(2t+1$$)^{10}$$}$$

Question 6

A medication's concentration in the bloodstream is modeled by $$C(t) = e$$^{\sin t}$$, where t$ is time in hours. What is the rate of change of concentration with respect to time?

Accepted Answer

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

Question 7

Differentiate $$y = (2x + 1)^5$ (x^3$ - x + 1)^4$.

Accepted Answer

$$2(2x + 1)^4$(x^3 - x$$ + $$1)^3(17x^3$ + 6x^2$ - 9x + 3)$$

Question 8

Which of the following is the correct application of the Chain Rule in Leibniz notation for $$y = f(u)$$ and $$u = g(x)$$?

Accepted Answer

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

Question 9

Find the derivative of $$y = e$$^{\sec 3	heta}$$ with respect to $$	heta$$.

Accepted Answer

$$3e^{\sec 3	heta}$$ \sec 3	heta 	an 3	heta$$

Question 10

Suppose $$y = \sin^2$ x. $$What is $$\frac{dy}{dx}$$?

Accepted Answer

$$2\sin x \cos x$$

Question 11

A population grows according to $$P(t) = 5$$^{2t}$$. Find P$$'(t)$$.

Accepted Answer

$$2 \cdot 5$$^{2t}$$ \ln 5$$

Question 12

Which of the following is the correct derivative of $$F(x) = \sqrt{x^2$ + 1}$$?

Accepted Answer

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

Question 13

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

Accepted Answer

$$\frac{1}{2}(3x^2$ + 2x$$)^{-1/2}$$(6x + 2)$$

Question 14

Which of the following best explains why the Chain Rule is necessary for differentiating $$y = \sqrt{x^2$ + 1}$$?

Accepted Answer

$$y is a $$composite function, so we must differentiate the outer function and multiply by the derivative of the inner function.

Question 15

Find the derivative of $$y = \sin(2x).$$

Accepted Answer

$$2\cos(2x)$$

Question 16

If $$y = (5x - 4)^7$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$35(5x - 4)^6$

Question 17

A radioactive substance decays according to $$N(t) = 2$$^{1-3t}$$. What is N$$'(t)$$?

Accepted Answer

$$-3 \cdot 2$$^{1-3t}$$ \ln 2$$

Question 18

Which of the following is the correct generalization for the derivative of $$a^{g(x)}$$?

Accepted Answer

$$a^{g(x)}$$ \ln a \cdot g'(x)$$

Question 19

If $$y = \cos^4$(x)$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$4\cos^3$(x)(-\sin x)$$

Differentiation Rules: The Chain Rule and Its Applications

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