What is the nth derivative of x^2log(3x) ?

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

\[k^{th} derivative\ of\ the\ 3^{rd}\ derivative: \frac{(-1^k)2(k!)}{x^{k+1}} \]
\[
\therefore \frac{d^n(x^2\ln(3x))}{dx^n}=
\begin{cases}
2xln(3x)+x,&n=1\\
2ln(3x)+3,&n=2\\
2x^{-1},&n=3\\
\frac{(-1^k)2(k!)}{x^{k+1}},&k=n-3\ and n>3
\end{cases}
\]
\[ =
\begin{cases}
2xln(3x)+x,&n=1\\
2ln(3x)+3,&n=2\\
\frac{(-1^{n-3})2(n-3!)}{x^{n-2}},&n\geq3
\end{cases} \]

Also Read: Find the Derivative of 1/x, f(x) = 1/x

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