Let f(x) = 2 u * v where
u = sin (4x)
v= sin-1 (x) = arcsin (x)
Using the product and chain rules :
f ' (x) = 2 [ u ' * v + u * v ' ]
So
f ' (x) = 2 [ 4cos (4x)* arcsin (x) + sin (4x) * 1 / √ (1 - x^2 ) ] =
8cos(4x)arcsin (x) + 2 sin (4x) / √ (1 - x^2 )