\[\large \frac{d^n}{dx^n} \frac{\ln(x)}{x} = \frac{a_n\ln(x)-b_n}{x^{n+1}}\]

Let \(f^{(n)}(x)\) be defined as the \(n\)-th derivative of \(\frac{\ln(x)}{x}\).

If \(f^{(n)}(x)\) can be written in the form shown above, then the solution to \(f^{(n)}(x)=0\) can be written in the form \(x=e^{\frac{p_n}{q_n}}\) where \(p_n\) and \(q_n\) are coprime positive integers.

What is \(p_{10}+q_{10}\)?

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