y=(6x^9 +x^8)/(2x^4), work out the value of d^2y/dx^2 when x=0.5

The question can be represented by the notation d²y/dx²_|x=0.5, meaning the second derivative of y with respect to x resolved at x=0.5. Since y is in the form f(x)/g(x), the quotient rule could be used, but it would be much easier to first simplify y to 3x⁵ + x⁴/2, using the index rules (x^m/xⁿ = x^m-n). Once y is in this form we can easily differentiate both terms with respect to x twice, giving dy/dx = 15x⁴ + 2x³, and then d²y/dx² = 60x³ + 6x². At this point we can substitute in x=0.5, giving d²y/dx²_|x=0.5 = 60(0.5)³ + 6(0.5)² = 9.