Question 1180593
If {{{f(x)=8ln(6x+3ln(x))}}}. Find {{{f}}}'{{{(x)}}}



{{{(d/dx)(8ln(6x+3ln(x)) )}}}...........take constant out


={{{8(d/dx)(ln(6x+3ln(x)) )}}}..........apply the chain rule


={{{(8/(6x+3ln(x)))*(d/dx)*(6x+3ln(x))}}}


then {{{(d/dx)(6x+3ln(x) )=6+3/x}}}


={{{(8/(6x+3ln(x)))*(6+3/x)}}}.....simplify


={{{(8/(6x+3ln(x)))*((6x+3)/x)}}}


={{{(8/(3(2x+ln(x))))*(3(2x+1)/x)}}}


={{{(8/(cross(3)(2x+ln(x))))*(cross(3)(2x+1)/x)}}}



={{{(8(2x+1))/((2x+ln(x)*x))}}}


={{{(16x+8)/(2x^2+x*ln(x))}}}


so,

{{{f}}}'{{{(x) = (16x + 8)/(2x^2 + x*log(x))}}}