Question

What is the second derivative of `1/(1+x^2)`?

Answer 1

`d^((2))/(dx^2) (1/(1+x^2)) = frac (6x^2-2) ((1+x^2)^3)`

Explanation:

`f(x) = 1/(1+x^2)`

Based on the chain rule:

`f'(x) = d/(dx) (1+x^2)^(-1) =(2x)(-1)(1+x^2)^(-2) = (-2x)/((1+x^2)^2)`

Using the quotient rule:

`f''(x) = frac ((-2)(1+x^2)^2 +2x*2x *2(1+x^2)) ((1+x^2)^4) = frac ((-2)(1+x^2) +8x^2) ((1+x^2)^3)= frac (6x^2-2) ((1+x^2)^3)`