Question 1

Let $$h(x) = \sqrt{x^2$ - 2}. $$Using the chain rule, what is $$h'(x)$$?

Accepted Answer

$$h'(x) = \dfrac{x}{\sqrt{x^2$ - 2}}$$

Question 2

Given $$y = (2x + 1)^4$, what is $$\dfrac{dy}{dx}$$?

Accepted Answer

$$\dfrac{dy}{dx} = 8(2x + 1)^3$

Question 3

Suppose $$y = (2x^3$ + 2x)^6$. Using the chain rule, what is $$\dfrac{dy}{dx}$$?

Accepted Answer

$$\dfrac{dy}{dx} = 6(2x^3$ + 2x)^5$ \cdot (6x^2$ + 2)$$

Question 4

Find the equation of the tangent line to $$y = \sqrt{x^4$ + 8x} at $$the point $$(1, 3).$$

Accepted Answer

$$y = 2x + 1$$

Question 5

The revenue function for a product is $$R(x) = 100\sqrt{x^2$ + 31}$$, where $$x is $$the number of items sold. What is the rate at which revenue is changing when $$x = 15$$?

Accepted Answer

$$\dfrac{dR}{dx} = \dfrac{1500}{\sqrt{256}} = 93.75$$

Question 6

Given $$y = 2x^4$ - 7x^3$ + 1$$, what is the second derivative $$y''$$?

Accepted Answer

$$y'' = 24x^2$ - 42x$$

Question 7

If $$y = \dfrac{1}{x^2$}$$, what is the second derivative $$y''$$?

Accepted Answer

$$y'' = 6x$$^{-4}$$

Question 8

Let $$f(x) = 5\sqrt{x}. $$What is $$f''(x)$$?

Accepted Answer

$$f''(x) = -\dfrac{5}{4}x$$^{-3/2}$$

Question 9

Suppose the position of a particle is $$s(t) = t^3$ + t. $$What is the acceleration $$a(t)$$?

Accepted Answer

$$a(t) = 6t$$

Question 10

Given $$y = (x - 1)^4$, what is $$\dfrac{dy}{dx}$$?

Accepted Answer

$$\dfrac{dy}{dx} = 4(x - 1)^3$

Question 11

If $$y = \dfrac{3}{\sqrt{3x^2$ + 1}}$$, what is $$\dfrac{dy}{dx}$$?

Accepted Answer

$$\dfrac{dy}{dx} = -\dfrac{9x}{(3x^2$ + 1$$)^{3/2}$$}$$

Question 12

Given $$f(x) = \dfrac{4x^2$}{(7 - 5x)^3$}$$, find $$f'(x)$$ using the quotient and chain rules.

Accepted Answer

$$f'(x) = \dfrac{8x(7 - 5x)^3$ - 60x^2$(7 - 5x)^2$}{(7 - 5x)^4$}$$

Question 13

If $$y = 3x + \dfrac{1}{2x - 3}$$, what is $$y''$$?

Accepted Answer

$$y'' = \dfrac{6}{(2x - 3)^3$}$$

Question 14

A company's cost function is $$C(x) = 50x^2$ + 100x + 200. $$What is the marginal cost when $$x = 10$$?

Accepted Answer

$$C'(10) = 1100$$

Question 15

Given $$y = (x^3$ - 4x)^5$, what is $$\dfrac{dy}{dx}$$?

Accepted Answer

$$\dfrac{dy}{dx} = 5(x^3$ - 4x)^4$(3x^2 - 4$$)$

Question 16

If $$y = \left(\dfrac{3x - 1}{5x + 2}\$$right)^4$$, what is $$\dfrac{dy}{dx}$$?

Accepted Answer

$$\dfrac{dy}{dx} = 4\left(\dfrac{3x - 1}{5x + 2}\$$right)^3$$ \cdot \dfrac{13}{(5x + 2)^2$}$$

Question 17

Suppose $$s(t) = t^3$ + t is $$the position of a particle. What are the velocity and acceleration at $$t = 4$$ seconds?

Accepted Answer

Velocity $$= 49$$, Acceleration $$= 24$$

Chain Rule, Higher Order Derivatives, and Applications in Business Calculus

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