Doing integration yields-

then finally f(x)=-x cos x + sin x

To calculate derivative of the above function f(x) we have to apply formula of derivation of products of two functions-

F(x) =-x cos(x)+sin(x)

F’(x) =-[x(-sin x)+cos x.1]+cos x ;{by formula d/dx (f.g)=f.g’+g.f’}

F’(x) =-(-x sin x) - cos x + cos x

F’(x) = x.sin(x)