Question

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

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

Answer 1

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

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