Calculus: Early Transcendentals, Chapter 3, 3.2, Section 3.2, Problem 30

To find the derivative of the function f(x) = x/(x^2 - 1) use the quotient rule. The rule gives the derivative of f(x) = (g(x))/(h(x)) as f'(x) = (g'(x)*h(x) - g(x)*h'(x))/(h(x))^2
f(x) = x/(x^2-1)
f'(x) = (x'*(x^2 - 1) - x*(x^2 - 1)')/(x^2 - 1)^2
= (x^2 - 1 - 2x^2)/(x^2 - 1)^2
= (-x^2 - 1)/(x^2 - 1)^2
= -(x^2 + 1)/(x^2 - 1)^2
f''(x) = (-(x^2 + 1)/(x^2 - 1))'
= ((-x^2 - 1)'*(x^2 - 1)^2 + (x^2 +1)*((x^2 - 1)^2)')/(x^2 - 1)^4
= (-2x*(x^2 - 1)^2 + (x^2 + 1)*2*(x^2 - 1)*2x)/(x^2 - 1)^4
= (-2x*(x^2 - 1) + (x^2 + 1)*2*2x)/(x^2 - 1)^3
= (-2x^3 + 2x + 4x^3 + 4x)/(x^2 - 1)^3
= (2x^3 + 6x)/(x^2 - 1)^3