Find the derivative of the function. y = [x + (x + sin2(x))4]6

To find the derivative of the given function y = [x + (x + sin^2 (x))^4]^6, we use the Chain Rule (f(u(x))´ =  f´(u(x))·u´(x): 
    dy/dx = 6[x + (x+sin^2 (x))^4]^6-1 ⋅d/dx [(x + sin^2 (x))^4]
where we first differentiate the outermost function which is a sixth degree. In our given function, the outermost function is a sixth degree, then a fourth degree and finally a quadratic.

We differentiate each function and multiply them together:
     dy/dx =  6[x + (x+sin^2 (x))^4]^5 ⋅(1 + 4(x + sin^2 (x))^(4-1)) ⋅d/dx (x + sin^2 x)
     dy/dx = 6[x + (x+sin^2 (x))^4]^5 ⋅(1 + 4(x + sin^2 (x))^3) ⋅(1 + 2sinxcosx)

Since weknow that sin2x = 2sinxcosx,
     dy/dx = 6[x + (x+sin^2 (x))^4]^5 ⋅(1 + 4(x + sin^2 (x))^3) ⋅(1 + sin2x)