# 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.

• Please don't post handwritten mathematical formulas as pictures. Use MathJax instead. – Herr K. Jun 8 '19 at 18:53

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 $$\mu$$: $$\sum_{\omega \in \Omega} p(\omega) c(\omega) = \int_{\Omega} p(\omega) c(\omega) \,\mu(\mathrm d \omega).$$
This is puzzling to me since the derivative of a indefinite integral of $$c_t$$ should be $$c_t$$, not 1.