We have, #f(x)=xsqrt(3x-e^x)#.
Using the Product Rule, we get,
#:. f'(x)=x*d/dx{(3x-e^x)^(1/2)}+sqrt(3x-e^x)*d/dx{x}......(star)#.
Here, by the Chain Rule,
#d/dx{(3x-e^x)^(1/2)}=1/2*(3x-e^x)^(1/2-1)*d/dx{(3x-e^x)}#,
#=1/2*(3x-e^x)^(-1/2)*[d/dx{3x}-d/dx{e^x}]#,
#=1/(2sqrt(3x-e^x)){3*1-e^x}#.
#rArrd/dx{(3x-e^x)^(1/2)}=(3-e^x)/(2sqrt(3x-e^x))......(star_1)#.
Utilising #(star_1)" in "(star)#, we have,
#f'(x)=x*(3-e^x)/(2sqrt(3x-e^x))+(sqrt(3x-e^x))1#.
# ={(3x-xe^x)+2(3x-e^x)}/(2sqrt(3x-e^x))#.
# rArr f'(x)=(9x-xe^x-2e^x)/(2sqrt(3x-e^x))#.
Enjoy Maths.!
