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 (xm/xn = xm-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.

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