Finding Higher-Order Derivatives In Exercise, find the second derivative of the function.

f (x) = ln x/x^3 + x

d^2y/dx^2 = 2/x^2

Step-by-step explanation:

f (x) = y = In x/x^3 + x

Differentiating x/x^3 = [x^3 (1) - x (3x^2) ] / (x^3) ^2 = (x^3 - 3x^3) / x^6 = - 2x^3/x^6 = - 2/x^3

Assuming u = x/x^3

In x/x^3 = In u

Differentiating In u = 1/u = x^3/x = x^2

Differentiating x = 1

dy/dx = x^2 (-2/x^3) + 1 = - 2/x + 1

Differentiating - 2/x = 2/x^2

Differentiating a constant (1) = 0

d^2y/dx^2 = 2/x^2 + 0 = 2/x^2