Given that y=(4x+1)^3*sin(2x) , find dy/dx

y=(4x+1)^3*sin(2x) - this is a product of two functions of x. It can be rewritten as y = u(x)*v(x)   ; where u(x) = (4x+1)^3 and v(x) = sin(2x)

Using the product rule: dy/dx = u'(x)*v(x) + v'(x)*u(x) where the ' (prime) notation denotes the differential with respect to x

u'(x) = 34(4x+1)^2   and  v'(x) = 2*cos(2x)   using either substitution or simplification rules for both

Therefore, using product rule, dy/dx=[ 34(4x+1)^2 ] * [ sin(2x) ]  +  [ 2*cos(2x) ] * [ (4x+1)^3 ] 

which simplifies to: dy/dx = 2(4x+1)^3*cos(2x) + 12(4x+1)^2