# 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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