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.!
