Q: Complete the following derivative: $$\frac{d}{dx}\left[\frac{2\sec x}{6-\sec x}\right].$$

$$\frac{2\sec x \tan x (\sec x - 6)}{(6-\sec x)^2$}$$

Q: If $$f(\theta) = 2\sin \theta \cos \theta$$, what is $$f'(\theta)$$?

$$2\cos^2$ \theta - 2\sin^2$ \theta$$

Question 1

Given the function $$f(x) = x^2$ - 6x - 9$$, for which value(s) of $$x$$ does the tangent line to the curve have a slope of $$0$$?

Accepted Answer

$$x = 3$$

Question 2

For the function $$f(x) = 3x^2$ + 6x^2$ + 15x + 3$$, at which value(s) of $$x$$ does the slope of the tangent line equal $$15$$?

Accepted Answer

$$x = 0.5$$

Question 3

Find the derivative of $$f(x) = (6 - x^2$)(x^2 - 3x + 4$$)$.

Accepted Answer

$$-2x(x^2 - 3x + 4) + (6$$ - $$x^2)(2x - 3$$)$$

Question 4

Compute the derivative of $$f(x) = 2 \sin x \cos x.$$

Accepted Answer

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

Question 5

Suppose the position of an object moving horizontally after $$t$$ seconds is given by $$f(t) = 2t^3$ - 21t^2$ + 60t. $$What is the velocity function of the object?

Accepted Answer

$$v(t) = 6t^2$ - 42t + 60$$

Question 6

Given $$f(x) = 	an$$^{-1}$$(e^{4x}$$)$$, compute f$$'(x)$.

Accepted Answer

$$\frac{4e$$^{4x}$$}{1 + e$$^{8x}$$}$$

Question 7

Find $$\frac{dy}{dx}$$ for $$y = \sqrt{x^2$ + 8}.$$

Accepted Answer

$$\frac{x}{\sqrt{x^2$ + 8}}$$

Question 8

Use implicit differentiation to find $$\frac{dy}{dx}$$ for $$4x^2 + xy = 28$$ - $$3y^2.$$

Accepted Answer

$$\frac{8x + y}{6y - x}$$

Question 9

Compute the derivative of $$f(x) = \csc$$^{-1}$$(2x + 1).$$

Accepted Answer

$$\frac{2}{|2x+1|\sqrt{(2x+1)^2$-1}}$$

Question 10

The volume of a sphere is increasing at a rate of $$5\ \mathrm{cm}^3/\mathrm{sec}. $$Find the rate of change of its surface area when its volume is $$\frac{500\pi}{3}\ \mathrm{cm}^3.$$

Accepted Answer

$$40\pi\ \mathrm{cm}^2/\mathrm{min}$$

Question 11

Find $$\frac{d}{dx}(\ln(x^2$ + 2)).$$

Accepted Answer

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

Question 12

Compute the derivative of $$f(x) = \cos$$^{-1}$$(\sqrt{2x}).$$

Accepted Answer

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

Question 13

Given $$f(x) = \frac{7-2x}{5x-x^2$}$$, find $$f'(x).$$

Accepted Answer

$$\frac{-2(5x-x^2$)-(7-2x)(5-2x)}{(5x-x^2$)^2$$}$

Question 14

Find $$\frac{d}{dx}(5x\ln(15x)).$$

Accepted Answer

$$5\ln(15x) + 5$$

Question 15

Complete the following derivative: $$\frac{d}{dx}\left[\frac{2\sec x}{6-\sec x}\right].$$

Accepted Answer

$$\frac{2\sec x 	an x (\sec x - 6)}{(6-\sec x)^2$}$$

Question 16

If $$f(	heta) = 2\sin 	heta \cos 	heta$$, what is $$f'(	heta)$$?

Accepted Answer

$$2\cos^2$ 	heta - 2\sin^2$ 	heta$$

Question 17

Given $$f(x) = 3x^3$ + 6x^2$ + 15x + 3$$, find $$f'(x).$$

Accepted Answer

$$9x^2 + 12x + 15$

Question 18

Suppose the position of an object is $$f(t) = 2t^3$ - 21t^2$ + 60t. $$What is the acceleration function $$a(t)$$?

Accepted Answer

$$a(t) = 12t - 42$$

Question 19

Find the equation of the tangent line to the curve $$4x^2 + xy = 28$$ - $$3y^2 at $$the point $$(-1, 3). $$Which of the following is correct?

Accepted Answer

$$y - 3 = \frac{5}{17}(x + 1)$$

Calculus I: Differentiation, Continuity, and Applications Study Guide

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