Question 1160035
if {{{f(x) = 4sqrt(x)ln(x)}}} find {{{f}}}'{{{(x)}}} and then {{{f}}}'{{{(2)}}}=?

 

{{{f}}}'{{{(x) =(d/dx)( 4sqrt(x)ln(x))}}}........take constant out

 {{{f}}}'{{{(x) =4(d/dx)(sqrt(x)* ln(x))}}}

apply the Product Rule: {{{(f*g)}}}'={{{f}}}'*{{{g}}}+{{{f}}}* {{{g}}}'


{{{ f}}}'{{{(x) =4((d/dx)sqrt(x)* ln(x)+sqrt(x)*(d/dx)*ln(x))}}}


{{{(d/dx)sqrt(x)=1/(2sqrt(x))}}}

{{{(d/dx)ln(x)=1/x}}}


 {{{f}}}'{{{(x) =4(ln(x)/(2sqrt(x))+sqrt(x)/x)}}}


 {{{f}}}'{{{(x) =4(ln(x)/(2sqrt(x))+(sqrt(x))^2/(x*sqrt(x)))}}}


 {{{f}}}'{{{(x) =4(ln(x)/(2sqrt(x))+x/(x*sqrt(x)))}}}


 {{{f}}}'{{{(x) =4(ln(x)/(2sqrt(x))+2/(2sqrt(x)))}}}

{{{f}}}'{{{(x) =2(ln(x)/(sqrt(x))+2/(sqrt(x)))}}}

{{{f}}}'{{{(x) =(2(ln(x)+2))/sqrt(x)}}}


then  

{{{f}}}'{{{(2)=(2(ln(2)+2))/sqrt(2)}}}

{{{f}}}'{{{(2)=3.808685}}}