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)