Q: Find $$\frac{dy}{dx}$$ for $$y = \tan^3$ \theta.$$

$$3 \tan^2$ \theta \sec^2$ \theta$$

Question 1

Given $$y = x^3$ \cos x$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$3x^2$$ \cos x - $$x^3$$ \sin x$$

Question 2

Find $$\frac{dy}{dx} if$$ $$y = \frac{x}{\sin x}.$$

Accepted Answer

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

Question 3

If $$y = e$$^{x^5}$$ \sin x$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$e^{x^5}$$ $$(5x^4$$ \sin x + \cos x)$$

Question 4

Find $$\frac{dy}{dx}$$ for $$y = 	an^3$ 	heta.$$

Accepted Answer

$$3 	an^2$ 	heta \sec^2$ 	heta$$

Question 5

Given $$y = \ln(5x^2$ + 4)$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{10x}{5x^2$ + 4}$$

Question 6

Use logarithmic differentiation to find $$\frac{dy}{dx}$$ for $$y = \ln \left( \frac{x}{\sqrt{x+1}} \right).$$

Accepted Answer

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

Question 7

Suppose the cost to produce $$x$$ widgets is $$C(x) = -0.001x^3$ + 0.05x^2$ + 0.6x + 800. $$What is the marginal cost when $$x = 100$$?

Accepted Answer

$$C'(100) = 9$$

Question 8

Two planes leave the same airport at the same time. One flies north at 400 miles per hour, the other east at 300 miles per hour. How fast is the distance between them increasing after 2 hours?

Accepted Answer

$$500$$ miles per hour

Question 9

A right circular cylindrical tank with radius $$20 cm is $$being drained at $$25 cm$^3/sec. $$How fast is the height of the water in the tank dropping?

Accepted Answer

$$\frac{1}{16\pi} cm/sec$$

Question 10

Find the linearization $$L(x) of$$ $$f(x) = \cos(2x) at$$ $$x = \frac{\pi}{6}.$$

Accepted Answer

$$L(x) = \frac{1}{2} - \sqrt{3}(x - \frac{\pi}{6})$$

Question 11

If $$y = \arctan(5x)$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{5}{1 + 25x^2$}$$

Question 12

Find $$\frac{d^2$y}{dx^2$}$$ for $$y = x^2$ e^x.$$

Accepted Answer

$$(x^2 + 4x + 2)e^x$

Question 13

If $$y = \sin(5x^2$ + 4)$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$10x \cos(5x^2$ + 4)$$

Question 14

Use logarithmic differentiation to find $$\frac{dy}{dx}$$ for $$y = (x^3$ + 1)^5$.

Accepted Answer

$$15x^2$$ $$(x^3$$ + $$1)^4$$

Question 15

If $$y = 2$$^{x^2 + 1}$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$2^{x^2 + 1}$$ \cdot 2x \ln 2$$

Question 16

Given $$y = \arctan(5x)$$, what is the value of $$\frac{dy}{dx} at$$ $$x = 1$$?

Accepted Answer

$$\frac{5}{26}$$

Question 17

If $$y = \ln \left( \frac{x}{\sqrt{x+1}} \right)$$, for which $$x is$$ $$\frac{dy}{dx} = 0$$?

Accepted Answer

$$x = 2$$

Question 18

Suppose the cost function is $$C(x) = -0.001x^3$ + 0.05x^2$ + 0.6x + 800. $$What does the marginal cost represent in this context?

Accepted Answer

The approximate cost of producing one more widget at production level $$x$$

Calculus I: Differentiation Techniques and Applications Study Guide

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