Nth Root of N

During my son’s math lesson today we got on the subject of \sqrt[n]n, which I prefer to write as n^{1/n}. We made a table of a few obvious values and limits:

    \[ \begin{tabular}{|r|c|} \hline $n$ & $n^{1/n}$\\ \hline $0\leftarrow$ & $0\leftarrow$ \\ 1 & 1 \\ 2 & $\sqrt{2} \approx 1.414$ \\ 3 & $\sqrt[3]{3} \approx 1.442$ \\ 4 & $\sqrt{2} \approx 1.414$ \\ $\rightarrow \infty$ & $\rightarrow 1$ \\ \hline \end{tabular}\]

So the values rise from n=0 \cdots 2, peak somewhere in n=2 \cdots 4, and asymptotically drop to 1 after that. So the maximum is probably between 2 and 4, and I asked my son how we could find it. And he suggested we find where the curve is flat and the derivative is zero.

And that’s when I realized I’d forgotten know how to find \frac{d}{dx}(x^{1/x}). My son is only 11 and we’ve only touched on calculus, so he couldn’t help. I figure the first step is to express it as \frac{d}{dx}(e^\frac{ln(x)}{x}), but I don’t know where to go from there.

So we turn to the internet, and I read Wikipedia’s entry for the nth root algorithm, which doesn’t answer my question but is nice to read because it’s a little gem. And then we get to WolframAlpha, which tells us everything we want to know. See for yourself, the WolframAlpha page for \frac{d}{dx}(x^{1/x}), which says the derivative is x^\frac{1}{x}(\frac{1}{x^2} - \frac{log(x)}{x^2}), which is zero when \frac{1}{x^2}=\frac{log(x)}{x^2} or 1=log(x) or x=e.

So the maximum of \sqrt[n]n is \sqrt[e]e\approx 1.4447. Not very surprising, but nice to have it confirmed. (When I described this problem to my neighbor he said the max would be at e as soon as he heard it was between 2 and 4.)

But I still don’t know how to calculate that derivative, except by asking WolframAlpha. Which brings up the question, is the internet making me stupid by making things easy? Or is it making me smarter by taking care of drudgery? I’d like to say it makes me smarter, but I know that, once in a while, what appears to be drudgery at first turns out to be important, providing unexpected insight.


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Comments

11 Responses to “Nth Root of N”

  1. Pavel Holoborodko on June 8th, 2010 10:02 am

    Hello!

    Thank you for using QuickLaTeX!

    One small tip – you can place latex code inside ${-}$!..${-}$ tags if you want formula to be centered on the page horizontally (use it without “{-}” characters). QuickLaTeX will do that automatically. Like here:
    $$!
    S=\sqrt{p(p-a)(p-b)(p-c)}
    $$

  2. Neal on June 8th, 2010 10:32 am

    Thanks Pavel, great advice. I just edited the above post and centered the table. Works like a charm.

  3. Pat Podenski on June 27th, 2010 11:19 am

    In wolframalpha, just go down to the derivative section where it shows the solution and click on ‘show steps’ and then wolframalpha will show you how to use the chain rule

  4. Neal on June 28th, 2010 9:39 pm

    Thanks Pat, I didn’t notice the ‘show steps’ link before. The derivation is very clear.

  5. Hayabusa brother on July 3rd, 2010 11:19 pm

    My friend and I were arguing about this! Now I know that I was right. lol! Thanks for making me sure!

    Sent from my Android phone

  6. Peter Luthy on July 5th, 2010 10:08 pm

    Hi Neal,

    Another essentially equivalent way to make this derivation is to use the chain rule on the logarithm first: if f=f(x) is any positive function with a derivative, then d/dx(log f)=(1/f)f'(x). Thus f'(x)=f(x)*d/dx(log f).

    Even though this is basically equivalent to what they do at wolframalpha, this can sometimes be easier than using the chain rule: for expressions like x^{something} this method is basically a product-rule-type derivative instead of a chain-rule-type derivative. Sometimes one of these types is easier to work with than the other. It’s also a great exercise for young people, your son, say, to familiarize themselves with the logarithm, which is one of the most ubiquitous functions in all of mathematics. One might also ask questions like, what is the limit as x goes to zero of x^x or x^{1/x}? Since one has no direct way to compute such a limit easily, one uses your idea that x=e^{log x}, that one may move a limit inside the argument of a continuous function, and that one can use L’Hospital’s rule to compute the limit of expressions like xlogx and the like. Anyway the moral is that it’s good to keep the logarithm at arm’s reach when you’re doing calculus problems!

    Regards,

    Peter

  7. Neal on July 7th, 2010 3:43 pm

    Hayabusa, glad to help!

    Peter, wow, excellent comment. I’ve read it 4 times so far, and I think I get it now. Thanks for taking the time.

  8. bas latex on March 21st, 2011 9:19 pm

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  9. Jick on July 3rd, 2011 11:44 am

    You might enjoy what I call the Whammo Function, W(x), so called for its behavior for x 0. On the reals you can define it using logarithms, as usual.

  10. Idan on August 15th, 2011 5:24 am

    I ran across a problem in my calculus book,
    which asks to find the supremum of “nth root of n”
    for natural values. I think the upper bound in this case should be 3^(1/3),am i right ?

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