Q: A drug is administered intravenously to a patient. Let $$A(t) = 25 + 3t^2$ (in milligrams) be the amount of drug in the patient's body at time t$ hours after 6am. What is the rate of change of the amount at 1pm?

$$rac{dA}{dt} = 6t$$; ext{ at } t = 7, rac{dA}{dt} = 42$

Question 1

Given the function $$f(x) = (5x^2$ + 2)(x^2$ - 3x + 1)$$, what is $$f'(x)$$?

Accepted Answer

$$f'(x) = 10x(x^2$ - 3x + 1) + (5x^2$ + 2)(2x - 3)$$

Question 2

A drug is administered intravenously to a patient. Let $$A(t) = 25 + 3t^2$ (in milligrams) be the amount of drug in the patient's body at time t$ hours after 6am. What is the rate of change of the amount at 1pm?

Accepted Answer

$$rac{dA}{dt} = 6t$$; 	ext{ at } t = 7, rac{dA}{dt} = 42$

Question 3

Find the equation of the tangent line to the graph of $$g(x) = 3x^2$ + 2x at $$the point where $$x = 1.$$

Accepted Answer

$$y = 8x - 3$$

Question 4

Suppose $$f(x) = x^2$, g(x$$) = $$x^3$$, and $$h(x) = e$$^{x^2}$$g(x). $$Find $$h'(x).$$

Accepted Answer

$$h'(x) = e$$^{x^2}$$(3x^2$$ + $$2x^3$$)$

Question 5

If $$y = rac{e^x}{1 - 2e^x}$$, what is $$y'$$?

Accepted Answer

$$y' = rac{e^x}{(1 - 2e^x)^2$}$$

Question 6

Let $$h(x) = rac{f(x)}{4x + g(x)}$$ where $$f(5) = 10$$, $$g(5) = 2$$, $$f'(5) = 10$$, and $$g'(5) = -4. $$Find $$h'(5).$$

Accepted Answer

$$h'(5) = rac{10 	imes (-4) - 10 	imes (4 	imes 5 + 2)}{(4 	imes 5 + 2)^2$}$$

Question 7

Differentiate $$y = 	an(5x)$$ with respect to $$x.$$

Accepted Answer

$$y' = 5 	ext{sec}^2(5x)$$

Question 8

If $$h(x) = 	ext{ln}(4x^3$ + 5x^2$)$$, what is $$h'(x)$$?

Accepted Answer

$$h'(x) = rac{12x^2$ + 10x}{4x^3$ + 5x^2$}$$

Question 9

Find the derivative of $$y = 	ext{sin}(2x + 1)$$ with respect to $$x.$$

Accepted Answer

$$y' = 2	ext{cos}(2x + 1)$$

Question 10

Suppose $$f(x) = x^2$, g(x$$) = $$x^3$$, and $$h(x) = 3f(x)g(x). $$Find $$h'(x).$$

Accepted Answer

$$h'(x) = 3(2x 	imes x^3$ + x^2$ 	imes 3x^2$)$$

Question 11

If $$y = 	ext{ln}(x^2$ + 1)$$, what is $$y'$$?

Accepted Answer

$$y' = rac{2x}{x^2$ + 1}$$

Question 12

Find the derivative of $$y = 	ext{cos}(4x)$$ with respect to $$x.$$

Accepted Answer

$$y' = -4	ext{sin}(4x)$$

Question 13

If $$g(x) = rac{x^2$ - 5}{2x + 1}$$, what is $$g'(x)$$?

Accepted Answer

$$g'(x) = rac{2x(2x + 1) - (x^2$ - 5) 	imes 2}{(2x + 1)^2$}$$

Question 14

Suppose $$f(x) = 2x$$, $$g(x) = x^2$, and h(x) = f(x) + g(x$$)$$. What is h$$'(x)$$?

Accepted Answer

$$h'(x) = 2 + 2x$$

Question 15

If $$y = 	ext{sin}(x^3$ + 7)$$, what is $$y'$$?

Accepted Answer

$$y' = 3x^2$ 	ext{cos}(x^3$ + 7)$$

Question 16

Find the derivative of $$y = x^4$ + 3x^2$ + 10$$ with respect to $$x.$$

Accepted Answer

$$y' = 4x^3$ + 6x$$

Question 17

If $$y = 	ext{ln}(x^2$ - 3x + 1)$$, what is $$y'$$?

Accepted Answer

$$y' = rac{2x - 3}{x^2$ - 3x + 1}$$

Question 18

Suppose $$f(x) = x^2$, g(x$$) = $$x^3$$, and $$h(x) = f(x)g(x). $$What is $$h'(x)$$?

Accepted Answer

$$h'(x) = x^2$ 	imes 3x^2$ + x^3$ 	imes 2x$$

Differentiation and Applications: Practice Problems and Solutions

Study Guide - Practice Questions

Study Guide - Flashcards