Answer to Question #97224 in Differential Equations for Henry Attah

Question

Compute the n-th differential coefficient of \\(y=x\\log_{e}x\\)\n

Answer 1

$y=(x\ln x)^n$ ;

$y'=n(\ln x+1)(x\ln x)^{n-1}$ ;

$y''=n(n-1)(\ln x+1)(x\ln x)^{n-2}+n\frac{1}{x}(x\ln x)^{n-1}=n(x\ln x)^{n-1}\left(\frac{(n-1)(\ln x+1)}{x\ln x}+\frac{1}{x}\right)$ ;

$y^{(n)}=n!\ln^nx$