Q: Let $$y = \sin(\tan(x^2$)). $$What is $$\dfrac{dy}{dx}$$?

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

Question 1

Let $$f(x)$$ and $$g(x) be $$differentiable functions. If $$F(x) = f(g(x))$$, and at $$x = 2 we $$have $$g(2) = 3$$, $$g'(2) = -1$$, $$f(3) = 5$$, and $$f'(3) = 4$$, what is $$F'(2)$$?

Accepted Answer

$$-4$$

Question 2

Suppose $$y = \sin(\ln(x^2$)). $$What is $$\dfrac{dy}{dx}$$?

Accepted Answer

$$\dfrac{2}{x} \cos(\ln(x^2$))$$

Question 3

Let $$F(x) = \dfrac{f(x)}{g(f(x))}$$, where $$f(2) = 4$$, $$f'(2) = 3$$, $$g(4) = 2$$, and $$g'(4) = 2. $$What is $$F'(2)$$?

Accepted Answer

$$\dfrac{3 - 12}{2}$$

Question 4

Suppose $$y = 	an$$^{-1}$$(e^{x^2}$$)$$. What is $$\dfrac{dy}{dx}$?

Accepted Answer

$$\dfrac{2x e$$^{x^2}$$}{1 + e$$^{2x^2}$$}$$

Question 5

A particle moves along a coordinate axis so that its position is given by $$s(t) = 2\cos(t) + \sqrt{3} t. $$For $$0 \u003c t \u003c 2\pi$$, at what time(s) is the particle's acceleration equal to $$\sqrt{3}$$?

Accepted Answer

$$t = \dfrac{5\pi}{6}$$

Question 6

Let $$y = \ln(2x - (3 \ln x)^4$). $$Compute $$y'.$$

Accepted Answer

$$\dfrac{2 - 12(\ln x)^3$ / x}{2x - (3 \ln x)^4$}$$

Question 7

Suppose $$f(x) is $$differentiable and $$f(1) = 2$$, $$f'(1) = 3. $$Let $$F(x) = x^2$ f(x^2$). $$What is $$F'(1)$$?

Accepted Answer

$$2 \cdot 2 + 2 \cdot 3 = 10$$

Question 8

Let $$y = \sin(	an(x^2$)). $$What is $$\dfrac{dy}{dx}$$?

Accepted Answer

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

Question 9

Suppose $$y = \ln(\sin(2x)). $$What is $$\dfrac{dy}{dx}$$?

Accepted Answer

$$2 \cot(2x)$$

Question 10

A tank in the shape of an inverted cone with base radius 2 m and height 4 m is being filled with water at a rate of $$2 	ext{ m}^3/	ext{min}. $$When the water is 3 m deep, at what rate is the water level rising?

Accepted Answer

$$\dfrac{2}{9\pi} m/$$min

Question 11

Let $$y = 	an$$^{-1}$$(x). $$What is $$\dfrac{dy}{dx}$$?

Accepted Answer

$$\dfrac{1}{1 + x^2$}$$

Question 12

Suppose $$y = \cos(2x^3$). $$What is $$\dfrac{dy}{dx}$$?

Accepted Answer

$$-6x^2$$ \$$sin(2x^3$$)$$

Question 13

Let $$y = \ln(\ln(\ln x)). $$What is $$\dfrac{dy}{dx}$$?

Accepted Answer

$$\dfrac{1}{x \ln x \ln(\ln x)}$$

Question 14

A dolphin's position is given by $$P(t) = e$$^{-t}$$ \sin\left(\dfrac{\pi}{2} - t\right). $$What is the speed of the dolphin at $$t = 0$$?

Accepted Answer

$$1 - \dfrac{\pi}{2}$$

Question 15

Let $$y = \sin$$^{-1}$$(x^2$$)$$. What is $$\dfrac{dy}{dx}$?

Accepted Answer

$$\dfrac{2x}{\sqrt{1 - x^4$}}$$

Question 16

Suppose $$y = (1 + x^2$)^{1/3}. $$What is $$\dfrac{dy}{dx}$?

Accepted Answer

$$\dfrac{2x}{3(1 + x^2$)^{2/3}$$}$

Question 17

Let $$y = 	an(\sin(x^2$)). $$What is $$\dfrac{dy}{dx}$$?

Accepted Answer

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

Question 18

Suppose $$y = \ln(	an(x)). $$What is $$\dfrac{dy}{dx}$$?

Accepted Answer

$$\sec^2$(x) / 	an(x)$$

Question 19

Let $$y = \sin(\cos(x)). $$What is $$\dfrac{dy}{dx}$$?

Accepted Answer

$$-\sin(x) \cos(\cos(x))$$

Question 20

Suppose $$y = \ln(\sin(2x)). $$What is $$\dfrac{dy}{dx}$$?

Accepted Answer

$$2 \cot(2x)$$

Calculus Exam 2 Study Guide: Derivatives, Chain Rule, Implicit Differentiation, and Related Rates

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