Answer :

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 :



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