Find the second order derivatives of the functions given.
`tan^(-1)x`.

Explanation

Let `y=tan^(-1)x`
Differentiating w.r.t. x

`dy/dx=(d(tan^(-1)x))/dx`

`dy/dx=1/(1+x^2)`

Again differentiating w.r.t. x

`d/dx(dy/dx)=d/dx(1/(1+x^2))`

`(d^2y)/(dx^2)=d/dx(1/(1+x^2))`

Using Quotient Rule
As, `u/v=(u'v-v'u)/v^2`
where `u=1` & `v=1+x^2`

`(d^2y)/(dx^2)=((d(1))/dx(1+x^2)-((d(1+x^2))/dx).1)/((1+x^2)^2)`

`(d^2y)/(dx^2)=(0*(1+x^2)-((d(1))/dx+(d(x^2))/dx)*1)/((1+x^2)^2)`

`(d^2y)/(dx^2)=(0-(0+2x)*1)/((1+x^2)^2)`

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