Lower The 6th Degree

Calculus Level 3

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

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

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

What is p10+q10p_{10}+q_{10}?

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