the derivative of 4xsin(x)cos(x)

Answer 1

using the product rule,
d/dx (4x)(sinx cosx)
= (4)(sinx cosx) + (4x) d/dx (sinx)(cosx)
= 4sinx cosx + 4x(cos^2x - sin^2x)
= 4(sinx cosx + x (cos^2x - sin^2x))

or, for less work,

4x sinx cosx = 2x sin2x
so the derivative is
2 sin2x + 4x cos2x