Q: Find $$\frac{d}{dx}(\tan(x + 10)^2$)$$ using the chain rule.

$$2(\tan(x + 10))(\sec^2$(x + 10))$$

Question 1

A stone is thrown vertically upward with an initial velocity of $$64 ft/s $$from a bridge $$16 ft $$above a river. Its position after $$t$$ seconds is given by $$s(t) = -16t^2$ + 64t + 16. $$What is the velocity of the stone when it reaches its maximum height?

Accepted Answer

$$0 ft/s$$

Question 2

Which of the following is the correct derivative of $$f(x) = 	an$$^{-1}$$(x)$$?

Accepted Answer

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

Question 3

Find $$\frac{dy}{dx} by $$implicit differentiation for the equation $$x^2$$ + $$y^2 = 25$.

Accepted Answer

$$\frac{dy}{dx} = -\frac{x}{y}$$

Question 4

A balloon is rising at a rate of $$4 m/s. A $$bicyclist passes beneath it, traveling at $$10 m/s. $$How fast is the distance between the bicyclist and the balloon increasing $$3$$ seconds later?

Accepted Answer

$$\frac{dD}{dt} = \frac{10x + 4y}{\sqrt{x^2$ + y^2$}}$$ evaluated at $$x = 30$$, $$y = 12$$; so $$\frac{dD}{dt} = \frac{10 	imes 30 + 4 	imes 12}{\sqrt{30^2$ + 12^2$}} = \frac{300 + 48}{\sqrt{900 + 144}} = \frac{348}{\sqrt{1044}} \approx 10.8 m/s$$

Question 5

Which rule is used to find the derivative of $$f(x) = e$$^{x^2}$$?

Accepted Answer

Chain Rule

Question 6

If $$y = x^x$$, what is $$\frac{dy}{dx}$$ using logarithmic differentiation?

Accepted Answer

$$x^x (\ln x + 1)$$

Question 7

Find the equation of the tangent line to $$y = x^2$ + y^2$ at the point (2$$, 3)$$.

Accepted Answer

$$y - 3 = (2x + 6)(x - 2)$$

Question 8

Which of the following is the correct derivative of $$f(x) = \ln(x^2$ + 1)$$?

Accepted Answer

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

Question 9

A ladder $$10 ft $$long rests against a vertical wall. If the bottom slides away from the wall at $$2 ft/s$$, how fast is the top sliding down the wall when the bottom is $$6 ft $$from the wall?

Accepted Answer

$$-1.5 ft/s$$

Question 10

Which formula gives the derivative of $$f(x) = \sin(x)$$?

Accepted Answer

$$\cos(x)$$

Question 11

Find $$\frac{d}{dx}(	an(x + 10)^2$)$$ using the chain rule.

Accepted Answer

$$2(	an(x + 10))(\sec^2$(x + 10))$$

Question 12

If $$f(x) = x^3$ and g(x$$) = \sin(x)$$, what is $$\frac{d}{dx}(f(x)g(x))$$?

Accepted Answer

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

Question 13

Which of the following is the correct derivative of $$f(x) = \sec(x)$$?

Accepted Answer

$$\sec(x) 	an(x)$$

Question 14

Find $$\frac{d}{dx}(e$$^{3x}$$).$$

Accepted Answer

$$3e^{3x}$$

Question 15

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

Accepted Answer

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

Question 16

A spherical balloon is being inflated so that its radius increases at a rate of $$2 cm/$$min. How fast is the volume increasing when the radius is $$5 cm$$? (Recall $$V = \frac{4}{3}\pi r^3$)

Accepted Answer

$$\frac{dV}{dt} = 4\pi r^2$ \frac{dr}{dt} = 4\pi (5)^2$ (2) = 200\pi cm$^3/$$min

Question 17

Which of the following is the correct derivative of $$f(x) = \ln(x)$$?

Accepted Answer

$$\frac{1}{x}$$

Question 18

If $$f(x) = x^2$ and g(x$$) = $$x^3$$, what is $$\frac{d}{dx}(f(x) + g(x))$$?

Accepted Answer

$$2x + 3x^2$

Question 19

Find $$\frac{d}{dx}(\sin(x) \cos(x)).$$

Accepted Answer

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

Calculus Exam 2 Study Guide: Derivatives, Implicit Differentiation, Tangent Lines, Logarithmic Differentiation, and Related Rates

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