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]

∴ [∵ ]

∴