Dixit Stiglitz price index FOC? In the solution to the problem below provided by my professor he applies a first order condition to the lagrangian for ct, getting pt for the partial derivative of the argument "$\frac{d}{dc_t(t)} \int p_t(w)*c_t(w)dw$". This is puzzling to me since the derivative of a indefinite integral of ct should be ct not 1. 

What's going on with this derivative and is it correct?

$\frac{d}{dc_t(t)} \int p_t(w)*c_t(w)dw$ = $p_t(w)$



Edit: Removed hand written notes.
A good way to think about this is to think about the cases where there are a finite number of goods, indexed by $\omega \in \Omega$. In this case, the partial derivative is
$$
p(\omega) = \frac{\partial}{\partial c(\omega)} \sum_{\omega' \in \Omega} p(\omega') c(\omega').
$$
The case in which $\Omega$ is a continuum of goods follows as a sort of limiting extension. See that even this finite sum can be expressed as an integral, given the proper measure $\mu$:
$$
\sum_{\omega \in \Omega} p(\omega) c(\omega) = \int_{\Omega} p(\omega) c(\omega) \,\mu(\mathrm d \omega).
$$

You mention,


  This is puzzling to me since the derivative of a indefinite integral
  of $c_t$ should be $c_t$, not 1.


If you provided your reasoning for this, it would be easier to see where the confusion is. In any case, hope this helps! 

Also, this guide may provide further assistance: The basics of “Dixit-Stiglitz lite”