Question 1

Given the function $$f(x) = x^3$ \sin(x)$$, what is $$f'(x)$$?

Accepted Answer

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

Question 2

Which of the following is the correct derivative of $$	an(x)$$?

Accepted Answer

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

Question 3

A ball is thrown vertically upward with an initial velocity of $$40 ft/s $$from a height of $$5 ft. $$Using $$h(t) = -16t^2$ + vt + h_0$, what is the maximum height reached by the ball?

Accepted Answer

$$55 ft$$

Question 4

If $$y = \sin(2x)$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$2\cos(2x)$$

Question 5

Which of the following is the correct application of the quotient rule for $$f(x) = \frac{x^2$}{\sin(x)}$$?

Accepted Answer

$$\frac{2x\sin(x) - x^2$\cos(x)}{\sin^2$(x)}$$

Question 6

Evaluate $$\lim_{x 	o 0} \frac{\sin(x)}{x}$$ using L'Hôpital's Rule.

Accepted Answer

$$1$$

Question 7

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

Accepted Answer

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

Question 8

Given $$f(x) = \ln(x^2$ + 1)$$, what is $$f'(x)$$?

Accepted Answer

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

Question 9

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

Accepted Answer

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

Question 10

A car's position is given by $$s(t) = 3t^2$ + 2t + 1. $$What is the instantaneous velocity at $$t = 4$$ seconds?

Accepted Answer

$$26$$ units/s

Question 11

Which of the following is the correct form of the Pythagorean identity?

Accepted Answer

$$\sin^2$(x) + \cos^2$(x) = 1$$

Question 12

If $$y = 	an$$^{-1}$$(x)$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

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

Question 13

Which of the following is the correct double angle identity for $$\sin(2x)$$?

Accepted Answer

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

Question 14

Given $$f(x) = e$$^{2x}$$, what is f$$'(x)$$?

Accepted Answer

$$2e^{2x}$$

Question 15

If $$f(x) = \frac{1}{x}$$, what is $$f'(x)$$?

Accepted Answer

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

Question 16

A ladder 20 feet long is leaning against a wall. If the bottom of the ladder slides away from the wall at a rate of 2 ft/s, how fast is the top of the ladder sliding down the wall when the bottom is 12 ft from the wall?

Accepted Answer

$$-1.5 ft/s$$

Question 17

Which of the following is the correct formula for the derivative of $$\cos(x)$$?

Accepted Answer

$$-\sin(x)$$

Question 18

If $$f(x) = (3x^2$ + 2x$$)^{10}$$, what is f$$'(x)$$?

Accepted Answer

$$10(3x^2$$ + $$2x)^9$$ \cdot (6x + 2)$$

Question 19

Find the equation of the tangent line to $$y = x^2$ - 3x + 2 at $$the point $$(2, 0).$$

Accepted Answer

$$y = x - 2$$

Question 20

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

Accepted Answer

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

Calculus Study Guide: Derivatives, Rules, and Applications

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