To perform the differentiation, let's analyze each term of the function.
Our function is y = 4x^3 + 3*9x^2 - 8.
1. For the first term, 4x^3, we use the power rule of differentiation, which states that the derivative of x^n = n*x^(n-1). We apply this rule to 4x^3 which gives us 3*4*x^(3-1) = 12x^2.
2. Next, we differentiate the second term, 3*9x^2. Again, we apply the power rule, getting 2*3*9*x^(2-1) = 54x.
3. Finally, when we differentiate the last term, -8, it becomes zero since a constant's derivative is always zero.
So, after differentiating each term, we have y' = 12x^2 + 54x, which is the derivative of the given function.
Therefore, y' = 12x^2 + 54x