Question

How do you differentiate `y= x+((x+sin^2x)^3) ^4`?

Answer 1

`y' = 1+12(x + sin^2(x))^11 (1-2sin(x)cos(x))`

Explanation:

This problem is solved using the chain rule:
`d/dx f(g(x)) = f'(g(x))*g'(x)`

`y = x + ((x + sin^2(x))^3)^4 = x + (x+sin^2(x))^12`

Taking the derivative:
`(dy)/dx = d/dx x + d/dx (x+sin^2(x))^12`
`= 1 + 12(x + sin^2(x))^11*(d/dx(x + sin^2(x)))`
`= 1 + 12(x + sin^2(x))^11*(d/dx x + d/dx sin^2(x))`
`= 1 + 12(x + sin^2(x))^11*(1 + 2sin(x)(d/dx sin(x)))`
`= 1 + 12(x + sin^2(x))^11(1 - 2sin(x)cos(x))`