# Lower The 6th Degree

Calculus Level 3

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

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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