Question

Y`=((1+sin2x)cos2x)`

Answer 1

D(y): sin2x<>0(знак <> означает "не равно")

2x<>пk, k-целое число.

x<>пk/2, k-целое число

Answer 2

To find the derivative of the function Y = (1 + sin(2x))cos(2x), we'll use the product rule and chain rule. The product rule states that if we have a function u(x) multiplied by another function v(x), then the derivative of their product is given by:

(d/dx)(u(x) * v(x)) = u'(x) * v(x) + u(x) * v'(x)

Now, let's proceed with finding the derivative step by step:

1. Let u(x) = 1 + sin(2x) and v(x) = cos(2x).

2. Find the derivatives of u(x) and v(x):

   u'(x) = d/dx(1 + sin(2x)) = 0 + cos(2x) * d/dx(sin(2x)) = cos(2x) * 2 * cos(2x) = 2cos^2(2x)

   v'(x) = d/dx(cos(2x)) = -sin(2x) * d/dx(2x) = -2sin(2x)

3. Now apply the product rule:

   Y' = u'(x) * v(x) + u(x) * v'(x) = (2cos^2(2x)) * cos(2x) + (1 + sin(2x)) * (-2sin(2x)) = 2cos^3(2x) - 2sin(2x) - 2sin(2x) * (1 + sin(2x))

Thus, the derivative of Y with respect to x is:

Y' = 2cos^3(2x) - 4sin(2x) - 2sin(2x) * sin(2x + 2)

Simplifying further if needed, but leaving it in this form is also acceptable.