\[1^{st} derivative: 2xln(3x)+x\]

\[2^{nd} derivative: 2ln(3x)+3\]

\[3^{rd}\ derivative: 2x^{-1}\]

\[k^{th} derivative\ of\ the\ 3^{rd}\ derivative: \frac{(-1^k)2(k!)}{x^{k+1}} \]

\[

\therefore \frac{d^n(x^2\ln(3x))}{dx^n}=

\begin{cases}

2xln(3x)+x,&n=1\\

2ln(3x)+3,&n=2\\

2x^{-1},&n=3\\

\frac{(-1^k)2(k!)}{x^{k+1}},&k=n-3\ and n>3

\end{cases}

\]

\[ =

\begin{cases}

2xln(3x)+x,&n=1\\

2ln(3x)+3,&n=2\\

\frac{(-1^{n-3})2(n-3!)}{x^{n-2}},&n\geq3

\end{cases} \]

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