$$y' = \frac{x^2$ \frac{d}{dx}(\tan x) - \tan x \frac{d}{dx}(x^2$)}{x^4$}$$

$$y'' = \frac{(x^2$ \frac{d^2$}{dx^2$}(\tan x) - 2x \frac{d}{dx}(\tan x) - \tan x \frac{d^2$}{dx^2$}(x^2$))}{x^4$}$$

$$x$$ where $$x^2$$ \frac{d}{dx}(\tan x) - \tan x \frac{d}{dx}$$(x^2) = 0$

$$y' = \frac{x^2$ \frac{d}{dx}(\tan x) - \tan x \frac{d}{dx}(x^2$)}{x^4$}$$

$$y' = \frac{1^2$ \frac{d}{dx}(\tan 1) - \tan 1 \frac{d}{dx}(1^2$)}{1^4$}$$

$$y'' = \frac{(x^2$ \frac{d^2$}{dx^2$}(\tan x) - 2x \frac{d}{dx}(\tan x) - \tan x \frac{d^2$}{dx^2$}(x^2$))}{x^4$}$$

$$y' = \frac{x^2$ \frac{d}{dx}(\tan x) - \tan x \frac{d}{dx}(x^2$)}{x^4$}$$

$$y'' = \frac{(x^2$ \frac{d^2$}{dx^2$}(\tan x) - 2x \frac{d}{dx}(\tan x) - \tan x \frac{d^2$}{dx^2$}(x^2$))}{x^4$}$$

$$y' = \frac{4 \frac{d}{dx}(\tan 2) - \tan 2 \frac{d}{dx}(4)}{16}$$

Question 1

Given $$y = \frac{	an x}{x^2$}$$, what is the first derivative $$y'$$?

Accepted Answer

$$y' = \frac{x^2$ \frac{d}{dx}(	an x) - 	an x \frac{d}{dx}(x^2$)}{x^4$}$$

Question 2

If $$y = \frac{	an x}{x^2$}$$, which of the following is the correct expression for the second derivative $$y''$$?

Accepted Answer

$$y'' = \frac{(x^2$ \frac{d^2$}{dx^2$}(	an x) - 2x \frac{d}{dx}(	an x) - 	an x \frac{d^2$}{dx^2$}(x^2$))}{x^4$}$$

Question 3

Given $$y = \frac{	an x}{x^2$}$$, which of the following best describes the behavior of $$y as$$ $$x 	o 0$$?

Accepted Answer

$$y$$ approaches infinity

Question 4

For $$y = \frac{	an x}{x^2$}$$, which of the following is a critical point?

Accepted Answer

$$x$$ where $$x^2$$ \frac{d}{dx}(	an x) - 	an x \frac{d}{dx}$$(x^2) = 0$

Question 5

Given $$y = \frac{	an x}{x^2$}$$, which of the following is the correct application of the quotient rule?

Accepted Answer

$$y' = \frac{x^2$ \frac{d}{dx}(	an x) - 	an x \frac{d}{dx}(x^2$)}{x^4$}$$

Question 6

Given $$y = \frac{	an x}{x^2$}$$, what is the value of $$y' at$$ $$x = 1$$?

Accepted Answer

$$y' = \frac{1^2$ \frac{d}{dx}(	an 1) - 	an 1 \frac{d}{dx}(1^2$)}{1^4$}$$

Question 7

Given $$y = \frac{	an x}{x^2$}$$, which of the following is the correct expression for $$y''$$ using the quotient rule?

Accepted Answer

$$y'' = \frac{(x^2$ \frac{d^2$}{dx^2$}(	an x) - 2x \frac{d}{dx}(	an x) - 	an x \frac{d^2$}{dx^2$}(x^2$))}{x^4$}$$

Question 8

Given $$y = \frac{	an x}{x^2$}$$, which of the following is the correct expression for $$y' in $$terms of $$	an x$$ and $$x$$?

Accepted Answer

$$y' = \frac{x^2$ \frac{d}{dx}(	an x) - 	an x \frac{d}{dx}(x^2$)}{x^4$}$$

Question 9

Given $$y = \frac{	an x}{x^2$}$$, which of the following is the correct expression for $$y'' in $$terms of $$	an x$$ and $$x$$?

Accepted Answer

$$y'' = \frac{(x^2$ \frac{d^2$}{dx^2$}(	an x) - 2x \frac{d}{dx}(	an x) - 	an x \frac{d^2$}{dx^2$}(x^2$))}{x^4$}$$

Question 10

Given $$y = \frac{	an x}{x^2$}$$, which of the following is the correct expression for $$y' at$$ $$x = 2$$?

Accepted Answer

$$y' = \frac{4 \frac{d}{dx}(	an 2) - 	an 2 \frac{d}{dx}(4)}{16}$$

Advanced Differentiation Techniques in Calculus: Logarithmic and Exponential Functions

Study Guide - Practice Questions