Question

first principle to compute the derivative of xsin2x

Answer 1

u = x, v=sin(2x)
u' * v + u * v'

Answer 2

using product rule to compute derivative of function F(x) = u v

it states f'(x) = (u)(derivative of v) + (v)(derivative of u)

now in order to find derivative of x sin(2x) , use above rule

that is, F'(x) = (x)(derivative of sin 2x) + (sin2x)(derivative of x)
= (x)(2cos2x) + (sin2x) (1) = 2xcos2x + sin 2x