Q: Given $$y^2$$ \\cos y = $$x^2 - x$$ \\cos x$$, find $$\\frac{dy}{dx}$$ using implicit differentiation.

$$\\frac{dy}{dx} = \\frac{2x - \\cos x - x \\sin x}{2y - \\cos y - y \\sin y}$$

Q: If $$f(x) = \\tan(x + 10)^2$, what is $$\\frac{d}{dx}f(x)$$?

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

Q: A balloon is rising at a rate of 5 m/s from a point 100 m away from an observer. How fast is the distance between the observer and the balloon increasing when the balloon is 50 m above the ground?

$$\\frac{dD}{dt} = \\frac{5 \\times 100}{\\sqrt{100^2$ + 50^2$}} = \\frac{500}{\\sqrt{12500}} = \\frac{500}{111.8} \\approx 4.47 m/s$$

Q: Find the equation of the tangent line to the curve $$y^2$$ \\cos y = $$x^2 - x$$ \\cos x$$ at the point (2$$, 3)$$.

$$y - 3 = \\frac{2 \\times 2 - \\cos 2 - 2 \\sin 2}{2 \\times 3 - \\cos 3 - 3 \\sin 3}(x - 2)$$

Q: 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?

$$\\frac{dy}{dt} = -\\frac{6}{8} \\times 2 = -1.5 ft/s$$

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

$$x^2$$ \\frac{1}{x} + 2x \\ln(x)$$

Q: Which of the following is the correct formula for the derivative of $$f(x) = \\arcsin(x)$$?

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

Q: A spherical balloon is being inflated so that its radius increases at a rate of 2 cm/s. How fast is the volume of the balloon increasing when the radius is 10 cm? (Recall $$V = \\frac{4}{3} \\pi r^3$)

$$\\frac{dV}{dt} = 4 \\pi r^2$ \\frac{dr}{dt} = 4 \\pi (10)^2$ \\times 2 = 800 \\pi cm$^3/s$$

Question 1

A stone is thrown vertically upward from a bridge 96 ft above a river with an initial velocity of 64 ft/s. Its position after $$t$$ seconds is given by $$s(t) = -16t^2$ + 64t + 96. $$What is the velocity of the stone when it reaches its maximum height?

Accepted Answer

$$v = 0$$

Question 2

Given $$y^2$$ \cos y = $$x^2 - x$$ \cos x$$, find $$\frac{dy}{dx}$$ using implicit differentiation.

Accepted Answer

$$\frac{dy}{dx} = \frac{2x - \cos x - x \sin x}{2y - \cos y - y \sin y}$$

Question 3

If $$f(x) = \tan(x + 10)^2$, what is $$\frac{d}{dx}f(x)$$?

Accepted Answer

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

Question 4

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

Accepted Answer

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

Question 5

A balloon is rising at a rate of 5 m/s from a point 100 m away from an observer. How fast is the distance between the observer and the balloon increasing when the balloon is 50 m above the ground?

Accepted Answer

$$\frac{dD}{dt} = \frac{5 \times 100}{\sqrt{100^2$ + 50^2$}} = \frac{500}{\sqrt{12500}} = \frac{500}{111.8} \approx 4.47 m/s$$

Question 6

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

Accepted Answer

$$y - 3 = \frac{2 \times 2 - \cos 2 - 2 \sin 2}{2 \times 3 - \cos 3 - 3 \sin 3}(x - 2)$$

Question 7

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

Accepted Answer

Chain rule

Question 8

If $$f(x) = x^2$ \sin x$$, what is $$f'(x)$$?

Accepted Answer

$$2x \sin x + x^2$ \cos x$$

Question 9

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

Accepted Answer

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

Question 10

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

$$\frac{dy}{dt} = -\frac{6}{8} \times 2 = -1.5 ft/s$$

Question 11

Which of the following is the correct formula for the derivative of $$f(x) = \sin(x)$$?

Accepted Answer

$$f'(x) = \cos(x)$$

Question 12

Find the derivative of $$f(x) = x^3$ + 2x^2$ - 5x + 7.$$

Accepted Answer

$$f'(x) = 3x^2$ + 4x - 5$$

Question 13

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

Accepted Answer

$$x^2$$ \frac{1}{x} + 2x \ln(x)$$

Question 14

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

Accepted Answer

$$f'(x) = \sec(x) \tan(x)$$

Question 15

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

Accepted Answer

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

Question 16

Find the second derivative of $$f(x) = x^3$.

Accepted Answer

$$f''(x) = 6x$$

Question 17

Which of the following is the correct formula for the derivative of $$f(x) = \arcsin(x)$$?

Accepted Answer

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

Question 18

A spherical balloon is being inflated so that its radius increases at a rate of 2 cm/s. How fast is the volume of the balloon increasing when the radius is 10 cm? (Recall $$V = \frac{4}{3} \pi r^3$)

Accepted Answer

$$\frac{dV}{dt} = 4 \pi r^2$ \frac{dr}{dt} = 4 \pi (10)^2$ \times 2 = 800 \pi cm$^3/s$$

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

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