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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.

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    $\begingroup$ How about computing a couple more? $\endgroup$ – Lord Shark the Unknown Oct 13 '18 at 18:55
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Hint:

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

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  • $\begingroup$ So this is true for $n\geq1$, which would be included in the conjecture, and I understand where you found this from because it is true. Am I supposed to try and prove that -$\frac{n+1}{x^{(n+1)+1}}$ is true to to prove the conjecture? $\endgroup$ – Claire Oct 13 '18 at 19:06
  • $\begingroup$ Maybe, depending on what you're supposed to know. Personally, I consider that every well-bred young people should know this formula by heart. $\endgroup$ – Bernard Oct 13 '18 at 19:12
  • $\begingroup$ I'm just supposed to be able to figure out the conjecture and prove it via induction $\endgroup$ – Claire Oct 13 '18 at 19:12
  • $\begingroup$ So I would do it in two step: first establish this formula (which is very easy) and next use it for the $n$-th derivative of log. $\endgroup$ – Bernard Oct 13 '18 at 19:16
  • $\begingroup$ You'll indeed obtain factorials, with a sign. Try first to formulate a conjecture, and test it for a few more orders of derivation, to see whether it seems valid, then prove it by induction. $\endgroup$ – Bernard Oct 13 '18 at 19:23
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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?

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  • $\begingroup$ $a_1$ would be $(n)!$ or (1)!, and then $a_{n+1}$ would be -(1)!, but how are you able to get to (6)! by the fourth derivative? $\endgroup$ – Claire Oct 13 '18 at 19:24
  • $\begingroup$ @Claire I think you meant $a_1=1=0!,\,a_{n+1}/a_n=n$. $\endgroup$ – J.G. Oct 13 '18 at 19:25
  • $\begingroup$ ohhhhh i see now! $\endgroup$ – Claire Oct 13 '18 at 19:27
  • $\begingroup$ okay so the conjecture could be $\frac{(-1)^{n-1}(n-1)!}{x^n}$ for n≥1? $\endgroup$ – Claire Oct 13 '18 at 19:31
  • $\begingroup$ @Claire So now prove it by induction. $\endgroup$ – J.G. Oct 13 '18 at 20:55

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