Q: Given $$f(x) = \tan x$$, what is $$\frac{d}{dx}(\tan x + 10$$)^{21}$$?

$$21(\tan x + 10$$)^{20}$$ \sec^2$ x$$

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 highest point?

Accepted Answer

$$0 ft/s$$

Question 2

Given $$f(x) = \ln(4x)$$, what is $$\frac{d}{dx}f(x)$$?

Accepted Answer

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

Question 3

If $$y = \sin$$^{-1}$$(x^2 - 1$$)$$, what is $$\frac{dy}{dx}$?

Accepted Answer

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

Question 4

For the function $$f(x) = 3x^3$ - 5x + 12$$, what is the third derivative $$f'''(x)$$?

Accepted Answer

$$18$$

Question 5

Suppose the radius $$r of an $$oil patch increases at a rate of $$30 m/hr. $$When the radius is $$100 m$$, how fast is the area increasing? (Area $$A = \pi r^2$)

Accepted Answer

$$6000\pi m$^2/hr$$

Question 6

Given $$y = x \ln x$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\ln x + 1$$

Question 7

Find the derivative of $$f(x) = e$$^{-3x}$$.

Accepted Answer

$$-3e^{-3x}$$

Question 8

If $$y = \csc x$$, what is $$\frac{d^2$y}{dx^2$}$$?

Accepted Answer

$$\csc x (\cot^2$ x + \csc^2$ x)$$

Question 9

A plane travels west at $$120 mi/hr $$and another north at $$150 mi/hr. At a $$certain instant, the westbound plane is $$180 mi $$and the northbound $$225 mi $$from the airport. How fast is the distance between them changing?

Accepted Answer

$$-192 mi/hr$$

Question 10

What is the derivative of $$f(x) = \sqrt{5x^2$ + 1}$$?

Accepted Answer

$$\frac{10x}{2\sqrt{5x^2$ + 1}}$$

Question 11

Given $$f(x) = 	an x$$, what is $$\frac{d}{dx}(	an x + 10$$)^{21}$$?

Accepted Answer

$$21(	an x + 10$$)^{20}$$ \sec^2$ x$$

Question 12

If $$y = \ln|\sec x|$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$	an x$$

Question 13

For $$f(x) = \frac{x^2$ + 1}{x^2$ - 4}$$, what is the slope of the tangent line at $$x = 3$$?

Accepted Answer

$$\frac{6}{5}$$

Question 14

What is the derivative of $$f(x) = \ln x^2$ / x^2$?

Accepted Answer

$$\frac{2(1 - \ln x^2$)}{x^3$}$$

Question 15

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

Accepted Answer

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

Question 16

What is logarithmic differentiation and when is it most useful?

Accepted Answer

It is a technique where you take the logarithm of both sides of a function before differentiating, and it is most useful for products, quotients, and powers of functions.

Question 17

Given $$f(x) = \frac{3x$$^{5/2}$$}{2x^2$ + 4}$$, what is $$\frac{d}{dx}f(x)$$?

Accepted Answer

$$\frac{15x$$^{3/2}$$(2x^2 + 4$$) - $$3x^{5/2}(4x$$)}{$$(2x^2$$ + $$4)^2$$}$

Question 18

Which of the following is the correct procedure for solving related rates problems?

Accepted Answer

Read the problem, write equations, differentiate, substitute known values, check units and reasonableness.

Comprehensive Study Guide: Derivatives and Their Applications (Exam 2, Calculus I)

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