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I'm getting confused when differentiating summations.

I understand that if you differentiate an expression and it gives a kroneker delta, that then sums over the appropriate summation and it disappears. But in my notes it has

[tex]\frac{\partial}{\partial p_{i}} [-k \sum_{i=1}^{r} p_{i} ln p_{i}][/tex]

Which I then differentiate to get

[tex]-k \sum_{i=1}^{r}[\frac{\partial p_{i}}{\partial p_{i}} ln p_{i} + p_{i} \frac{\partial}{\partial p_{i}} ln p_{i}][/tex]

Which I can get down to

[tex]-k \sum_{i=1}^{r}[ln p_{i} + 1][/tex]

But in the notes that summation sign has gone.....and I don't understand why. I know that if it had been...

[tex]\frac{\partial}{\partial p_{j}}[/tex] acting on it instead that the kroneker deltas would sum over the summation for me but it definitely has to be a subscript i.

Am I missing the point somewhere?

Thank you!

Hannah