Question

Using Leibnitz theorem find the nth derivative of y= x^2( log x)

Answer 1

See the solution in the attached file.

Answer 2

∴The required solution is $xy_{n+1} = y_{n}$

Step-by-step explanation:

Given equation is $y=x^{2} logx$

differentiating with respect to x both sides

⇒$\frac{dy}{dx}$= 2x㏒x +$x^{2} \frac{1}{x}$

⇒$\frac{dy}{dx}$=2x ㏒x+ x

⇒$\frac{dy}{dx}$=$2x\frac{y}{x^{2} }$+x

⇒$x\frac{dy}{dx}$=2y+x

⇒$xy_{1}$=2y+x

using leibnitz theorem

x$y_{n+1}$+$y_{n}$=2$y_{n}$

⇒$xy_{n+1} = 2y_{n} -y_{n}$

⇒$xy_{n+1} = y_{n}$