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I feel stupid for having to ask this, but does anybody have any idea how to handle $$\frac{d}{x}\sum_{n=k}^{g(x)}f(n,x)?$$ Example: $$\frac{d}{dx}\sum_{n=6}^{i^2+2i} \frac{1}{\ln{(i^2)}-\ln{\ln n}}.$$ If we were able to separate the summand into two functions, one with only $i$ as a variable, and one with only $n$ as a variable, this would be super simple. But it is not always the case. Any ideas?

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I take it $i$ in your sum was meant to be $x$. Think about a simpler example, like ${d\over dx}(1+2+\dots+x)$, and you may see what's happening. –  Gerry Myerson Mar 6 '11 at 23:03
    
@Gerry $1+2+\cdots+x$ is not a sum like the first one in this post. That is, unless $x$ is restricted to being an integer, and then I do not know the meaning of differentiation with respect to $x$. For $f(n,x)=n$ and $g(x)=x$, the resulting sum is $k+(k+1)+\cdots+\lfloor x\rfloor$. –  Did Mar 7 '11 at 6:29
    
Didier, yes, I was hoping Alex would work out what you've put forward. What does one make of ${d\over dx}(1+2+\cdots+[x])$? –  Gerry Myerson Mar 7 '11 at 6:34
    
@Gerry Then I should have kept quiet. Sorry. –  Did Mar 7 '11 at 11:04
    
Didier, I didn't mean to criticize you, and you have nothing to apologize for. But where has Alex gone? –  Gerry Myerson Mar 8 '11 at 12:10

1 Answer 1

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This reminds me uncomfortably about a remark that Terry Tao made in this answer about the importance of teaching the derivative as a limit: at least one person tried to prove Fermat's Last Theorem by differentiating with respect to the exponent. It does not make sense.

The reason why it does not: the derivative is something that is properly defined for real (or complex) variables. If you're variable is an integer, derivatives will not make sense.

There may objects that are analogous to the derivative in some ways, but you cannot carelessly apply derivative rules.

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