To find the derivative of the function f(x) = 0.5x^3 + 0.6x^2 + 0.8x + 8, you need to apply the power rule for differentiation to each term separately.

f'(x) = d/dx [0.5x^3] + d/dx [0.6x^2] + d/dx [0.8x] + d/dx [8]

f'(x) = 1.5x^2 + 1.2x + 0.8

Therefore, the derivative of the function f(x) = 0.5x^3 + 0.6x^2 + 0.8x + 8 is f'(x) = 1.5x^2 + 1.2x + 0.8.