Q: Find the derivative of $$y = x^2$ \tan x + \sqrt{x}.$$

$$y' = 2x \tan x + x^2$ \sec^2$ x + \dfrac{1}{2\sqrt{x}}$$

Q: Given $$y = \tan^2$(xe^x)$$, find $$\dfrac{dy}{dx}.$$

$$2 \tan(xe^x) \sec^2$(xe^x) (e^x + xe^x)$$

Q: Find the derivative of $$y = x^2$ \tan x - x^2$.

$$y' = 2x \tan x + x^2$ \sec^2$ x - 2x$$

Question 1

Let $$f(x) = \sqrt{3x+1}. $$Using the definition of the derivative, what is $$f'(x)$$?

Accepted Answer

$$f'(x) = \dfrac{3}{2\sqrt{3x+1}}$$

Question 2

Given $$f(x) = \sqrt{x^2$ + 1}$$, find the equation of the tangent line to the graph of $$f at$$ $$x = 7.$$

Accepted Answer

$$y = \dfrac{7}{\sqrt{50}}(x-7) + \sqrt{50}$$

Question 3

Suppose $$y = \dfrac{x}{\sqrt{x-a}}$$, where $$a is a $$positive constant. What is $$\dfrac{dy}{dx}$$?

Accepted Answer

$$\dfrac{dy}{dx} = \dfrac{\sqrt{x-a} + \dfrac{x}{2\sqrt{x-a}}}{x-a}$$

Question 4

A car traveling at $$48 ft/sec $$suddenly applies the brakes. The position of the car is $$s(t) = 48t - 3t^2$. How far does the car go before coming to a stop?

Accepted Answer

$$384 ft$$

Question 5

If $$y = e$$^{x^2}$$ \sin x^2$, find $$\dfrac{dy}{dx}$$.

Accepted Answer

$$2x e$$^{x^2}$$ \sin x^2$ + 2x e$$^{x^2}$$ \cos x^2$

Question 6

Find the second derivative of $$y = \dfrac{4}{x^2$}.$$

Accepted Answer

$$y'' = 24x$$^{-4}$$

Question 7

Given $$y = 	an(e$$^{\sqrt{3x}$$})$$, find $$\dfrac{dy}{dx}.$$

Accepted Answer

$$\sec^2$(e^{\sqrt{3x}$$}) \cdot $$e^{\sqrt{3x}$$} \cdot \dfrac{3}{2\sqrt{3x}}$

Question 8

The height $$h (in ft) of a $$stone above the ground $$t$$ seconds after it is thrown is $$h = -16t^2$ + 48t + 64. $$What is the velocity of the stone when it hits the ground?

Accepted Answer

$$-80 ft/sec$$

Question 9

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

Accepted Answer

$$f'''(x) = 1260x^4$ + 5e^x$$

Question 10

If $$y = \cos(5x^2$ + 11)$$, find $$\dfrac{dy}{dx}.$$

Accepted Answer

$$-10x \sin(5x^2$ + 11)$$

Question 11

Given $$y = 	an(x \cdot e^x)$$, find $$\dfrac{dy}{dx}.$$

Accepted Answer

$$\sec^2$(xe^x) \cdot (e^x + xe^x)$$

Question 12

Find the derivative of $$y = x^2$ 	an x + \sqrt{x}.$$

Accepted Answer

$$y' = 2x 	an x + x^2$ \sec^2$ x + \dfrac{1}{2\sqrt{x}}$$

Question 13

If $$y = (\cos(2x^7$))^4$$, find $$\dfrac{dy}{dx}$.

Accepted Answer

$$-56x^6$$ \$$sin(2x^7$$) \$$cos^3(2x^7$)$

Question 14

Find the derivative of $$y = \dfrac{e^x}{\sqrt{5}}.$$

Accepted Answer

$$y' = \dfrac{e^x}{\sqrt{5}}$$

Question 15

Given $$y = 	an^2$(xe^x)$$, find $$\dfrac{dy}{dx}.$$

Accepted Answer

$$2 	an(xe^x) \sec^2$(xe^x) (e^x + xe^x)$$

Question 16

If $$y = \cos(x^4$ + 12)$$, find $$\dfrac{dy}{dx}.$$

Accepted Answer

$$-4x^3$$ \$$sin(x^4 + 12$$)$$

Question 17

Find the derivative of $$y = x^2$ 	an x - x^2$.

Accepted Answer

$$y' = 2x 	an x + x^2$ \sec^2$ x - 2x$$

Question 18

If $$y = 	an(e$$^{x^2}$$)$$, find $$\dfrac{dy}{dx}.$$

Accepted Answer

$$\sec^2$(e^{x^2}$$) \cdot 2x $$e^{x^2}$

Question 19

Given $$y = 	an^2$(\sqrt{3x})$$, find $$\dfrac{dy}{dx}.$$

Accepted Answer

$$2 	an(\sqrt{3x}) \sec^2$(\sqrt{3x}) \cdot \dfrac{3}{2\sqrt{3x}}$$

Calculus Study Guide: Differentiation, Tangent Lines, and Applications

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