To find the derivative of Y = (1 + sin(2x))cos(2x), we will first simplify the expression by applying the product rule.

Y = (1 + sin(2x))cos(2x)
Y = cos(2x) + sin(2x)cos(2x)

Now, applying the product rule to the second term:

Y' = (-sin(2x) + cos(2x)cos(2x)) + (sin(2x)cos(2x))
Y' = -sin(2x) + cos^2(2x) + sin(2x)cos(2x)

Thus, the derivative of Y with respect to x is:
Y' = -sin(2x) + cos^2(2x) + sin(2x)cos(2x)