# Derivative of Sum over Variable of derivative

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