Find the second order derivatives of each of the following functions:

log (log 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 = log ( log x )

We have to find

As,

So lets first find dy/dx and differentiate it again.

∴

Let y = log t and t = log x

Using chain rule of differentiation:

∴ [∵ log x) = ]

Again differentiating w.r.t x:

As,

Where u = and v =

∴ using product rule of differentiation:

∴ [ use chain rule to find ]

[ ∵ ]