Question

Differentiate the following function with respect
to x.
Simplify your answer where possible.
y=(5x+2)^3 / (2x+1)

Answer 1

Huh, that's strange. Must've been my lousy internet not connecting properly. Anyhow, I'll just write the same thing.

This equation follows the form for the quotient rule:
\[y=\frac{ (5x+2)^3 }{ 2x+1 }=\frac{ g(x) }{ h(x) }\]
The quotient rule states:
\[y' =\frac{ h(x)g'(x) - h'(x)g(x) }{ [h(x)]^2 }\]
In this case, we would get:
\[y'=\frac{ (2x+1)[3(5x+2)^{2}(5)]-(2)(5x+2)^3 }{ (2x+1)^2 }\]\[y'=\frac{ (2x+1)(15(25x^2+20x+4))-(2(125x^3+150x^2+60x+8)) }{ (2x+1)^2 }\]\[y'=\frac{ (2x+1)(375x^2+300x+60)-(250x^3+300x^2+120x+16) }{ (2x+1)^2 }\]\[y'=\frac{ 750x^3+600x^2+120x+375x^2+300x+60-250x^3-300x^2-120x-16 }{ (2x+1)^2 }\]\[y'=\frac{ 500x^3+675x^2+300x+44 }{ (2x+1)^2 }\]This is where I would stop, but I do know this could be written slightly more condensed and nice as:
\[y'=\frac{ (5x+2)^2(20x+11) }{ (2x+1)^2 }\]

I would think either is fine.