quotient rule problem: y = (x^2) / (4 + 3x)

[letsgetaway]

For my problem I've got the correct written answer. I am confused how the answer is converted into the following answer choices. I was not able to figure it out. Anyone has a clue?

 

[galactus]

Check out answer a.

\(\displaystyle \L\\\frac{(4+3x)}{(4+3x)}\cdot\frac{2x}{4+3x}-\frac{3x^{2}}{(4+3x)^{2}}\)


\(\displaystyle \L\\\frac{8x+6x^{2}-3x^{2}}{(4+3x)^{2}}\)

\(\displaystyle \L\\\frac{3x^{2}+8x}{(4+3x)^{2}}\)


Here's what they done to get that:

Upon using the quotient rule, we get \(\displaystyle \L\\\frac{8x+6x^{2}-3x^{2}}{(4+3x)^{2}}\)

Now, instead of simplifying the usual way, they did this:

\(\displaystyle \L\\\frac{2x(4+3x)-3x^{2}}{(4+3x)^{2}}\)

\(\displaystyle \L\\\frac{2x(4+3x)}{(4+3x)^{2}}-\frac{3x^{2}}{(4+3x)^{2}}\)

\(\displaystyle \L\\\frac{2x}{4+3x}-\frac{3x^{2}}{(4+3x)^{2}}\)

 

[letsgetaway]

Thank you so much! I see what they are doing now. They didn't simplify it was much as I did. They just separated the quotient rule apart.

 

[galactus]

I did an edit since your last visit.

 

[letsgetaway]

Yup, I saw it.

 

[soroban]

Re: quotient rule problem

Hello, letsgetaway!

I know what they did . . . and it's pretty stupid!


They said: .\(\displaystyle y \:=\:\L\frac{x^2}{4\,+\,3x}\)\(\displaystyle \:=\:x^2\cdot(4\,+\,3x)^{-1}\)

and used the Product Rule:

. . \(\displaystyle y'\;=\;2x(4\,+\,3x)^{-1}\,+\,x^2\cdot(-1)(4\,+\,3x)^{-2}\cdot3 \;=\;\L\frac{2x}{4\,+\,3x}\,-\,\frac{3x^2}{(4\,+\,3x)^2}\)