Find the second order derivatives of each of the following functions:
tan-1 x
Basic idea:
√Second order derivative is nothing but derivative of derivative i.e. 
√The idea of chain rule of differentiation: If f is any real-valued function which is the composition of two functions u and v, i.e. f = v(u(x)). For the sake of simplicity just assume t = u(x)
Then f = v(t). By chain rule, we can write the derivative of f w.r.t to x as:
√Product rule of differentiation- 
√Apart from these remember the derivatives of some important functions like exponential, logarithmic, trigonometric etc..
Let’s solve now:
Given, y = tan –1 x
We have to find 
As 
So lets first find dy/dx and differentiate it again.
∴ 
[∵ 
tan–1 x) = 
]
∴ 
[∵ 
tan–1 x) = 
]
Differentiating again with respect to x :
Differentiating 
using chain rule,
let t = 1 +x2 and z = 1/t
∵ 
[ from chain rule of differentiation]
∴ 
[∵ 
]
∴ 