Basic Idea: Second order derivative is nothing but derivative of derivative i.e. \(\frac{d^2y}{dx^2}=\frac{d}{dx}(\frac{dy}{dx})\)
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:
\(\frac{df}{dx}=\frac{dv}{dt}\times\frac{dt}{dx}\)
Product rule of differentiation- \(\frac{d}{dx}(uv)=u\frac{dv}{dx}+v\frac{du}{dx}\)
Apart from these remember the derivatives of some important functions like exponential, logarithmic, trigonometric etc..
Let’s solve now:
As we have to prove : \(\frac{d^4y}{dx^4}=\frac{6}{x}\)
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
We have to find \(-\frac{d^4y}{dx^4}\)
So lets first find dy/dx and differentiate it again.
differentiating using product rule:
Again differentiating using product rule:
Again differentiating using product rule:
Again differentiating w.r.t x :
