<PRE><B>Question 1005760</B>
I&#39;m not sure why you introduced natural logs.


Here&#39;s how I would have done it


Let
{{{g(x) = (x+6)^8}}}
{{{h(x) = 4x-1}}}
{{{k(x) = (8x+6)^3}}}
{{{m(x) = g(x)*h(x) = (x+6)^8*(4x-1)}}}
{{{n(x) = (m(x))/(k(x))}}}


Based on those definitions, we know that


*[Tex \LARGE g \ &#39;(x) = 8(x+6)^7]


*[Tex \LARGE h \ &#39;(x) = 4]


*[Tex \LARGE k \ &#39;(x) = 24(8x+6)^2]


*[Tex \LARGE m \ &#39;(x) = g \ &#39;(x)*h(x) + g(x)*h \ &#39;(x)]


*[Tex \LARGE m \ &#39;(x) = 8(x+6)^7*(4x-1) + (x+6)^8*4]


*[Tex \LARGE m \ &#39;(x) = 4(x+6)^7\left[2(4x-1) + (x+6)\right]]


*[Tex \LARGE m \ &#39;(x) = 4(x+6)^7(8x-2 + x+6)]


*[Tex \LARGE m \ &#39;(x) = 4(x+6)^7(9x+4)]


I used the product rule to compute the derivative of m(x).


-------------------------------------------------


Then we compute the derivatie of n(x) using the quotient rule


*[Tex \LARGE n(x) = \frac{m(x)}{k(x)}]


*[Tex \LARGE n \ &#39;(x) = \frac{m \ &#39;(x)*k(x) - m(x)*k \ &#39;(x)}{\left[k(x)\right]^2}]


*[Tex \LARGE n \ &#39;(x) = \frac{4(x+6)^7(9x+4)*(8x+6)^3 - (x+6)^8*(4x-1)*24(8x+6)^2}{\left[(8x+6)^3\right]^2}]


*[Tex \LARGE n \ &#39;(x) = \frac{4(x+6)^7(8x+6)^2\left[(9x+4)(8x+6)-6(x+6)(4x-1)\right]}{(8x+6)^6}]


*[Tex \LARGE n \ &#39;(x) = \frac{4(x+6)^7(8x+6)^2\left[72x^2+86x+24-24x^2-138x+36\right]}{(8x+6)^6}]


*[Tex \LARGE n \ &#39;(x) = \frac{4(x+6)^7(8x+6)^2(48x^2-52x+60)}{(8x+6)^6}]


*[Tex \LARGE n \ &#39;(x) = \frac{4(x+6)^7(48x^2-52x+60)}{(8x+6)^4}]


-------------------------------------------------


Now onto the function f(x). Let&#39;s compute the derivative.


*[Tex \LARGE f(x) = \left(\frac{(x+6)^8(4x-1)}{(8x+6)^3}\right)^{\frac{1}{4}}]


*[Tex \LARGE f(x) = \left(n(x)\right)^{\frac{1}{4}}]


*[Tex \LARGE f \ &#39;(x) = \frac{1}{4}\left(n(x)\right)^{\frac{1}{4}-1}*\frac{d}{dx}(n(x))]


*[Tex \LARGE f \ &#39;(x) = \frac{1}{4}\left(n(x)\right)^{-\frac{3}{4}} * n \ &#39;(x) ]


*[Tex \LARGE f \ &#39;(x) = \frac{1}{4}\left(\frac{(x+6)^8(4x-1)}{(8x+6)^3}\right)^{-\frac{3}{4}} * \left[\frac{4(x+6)^7(48x^2-52x+60)}{(8x+6)^4}\right] ]

It&#39;s definitely a messy derivative, but it is the correct answer. I confirmed it with a graphing calculator and computer algebra system (CAS)</PRE>