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)