Conjecture for $\ln(x)$ and $\frac{d^ny}{dx^n}$

Question

I'm not really sure where to start, I found the first, second, third and fourth derivatives of $\ln(x)$ to be $\frac{1}{x}$,-$\frac{1}{x^2}$, $\frac{2}{x^3}$, and -$\frac{6}{x^4}$, respectively. Letting $n$ be a natural number, I have to formulate a conjecture for a formula for $\frac{d^ny}{dx^n}$. Afterward, I have to use mathematical induction to prove the conjecture.

Answer 1

You've probably at least guessed that a sequence $a_n$ of positive integers exists for which the $n$th derivative of $\ln x$ is $(-1)^{n-1}a_n/x^n$. What's $a_1$? What's $a_{n+1}$ in terms of $a_n$? Can you finish the solution using factorials?

Answer 2

Hint:

$$ \Bigl(\frac1{x^n}\Bigr)'=-\frac n{x^{n+1}}.$$