Y' = (arccos 3x^4)'(x+1) + (x+1)(arccos 3x^4)'

Using the chain rule:

Y' = (-1/sqrt(1-(3x^4)^2))(3x^4)'(x+1) + (x+1)(-1/sqrt(1-(3x^4)^2))(3x^4)'

Y' = (-1/sqrt(1-9x^8))(12x^3)(x+1) + (x+1)(-1/sqrt(1-9x^8))(12x^3)

Finally, simplifying the expression:

Y' = (-12x^3)(x+1)/sqrt(1-9x^8) - 12x^3(x+1)/sqrt(1-9x^8)