If y = x³ log x, prove that .

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:

As we have to prove :

We notice a third order derivative in the expression to be proved so first take the step to find the third order derivative.

Given, y = x³ log x

Let’s find -

As

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

∴

differentiating using product rule:

[log x) = ]

Again differentiating using product rule:

[log x) = ]

Again differentiating using product rule:

[log x) = ]

Again differentiating w.r.t x :