# Thread: Tricky differentiation w.r.t. to the index

1. ## Tricky differentiation w.r.t. to the index

$\displaystyle -\lambda\left(\int_0^1 P_t(i)C_t(i) di-Z_t\right)$

with respect to $\displaystyle C_t(i)$ using the standard rules of derivatives and integrals and get

$\displaystyle -\lambda P_t(i)$

i is an index representing firm i. There is a continuum of firms from 0 to 1. t is just an time index. The problem is from page 61 of Jordi Gali's book Monetary Policy, Inflation and the Business Cycles.

2. To put it in another more simple way, could you provide a formal proof of

$\displaystyle \frac{\partial }{\partial C(m)} \int_{0}^{1}P(i)C(i)di=P(m)$

I understand the intuition behind the result (I guess). One can think of the integral as an infinite sum, but I would like to see the math.

Anyone?

$\displaystyle \displaystyle \frac{\partial}{\partial C(m)} \int_0^1 \ P(i) \ C(i) \ di \ = \int_0^1 \frac{\partial}{\partial C(m)} \ P(i) \ C(i) \ di \ = \int_0^1 \ P(i) \ \frac{\partial}{\partial C(m)}\ C(i) \ di \ =$
$\displaystyle \displaystyle \int_0^1 \ P(i) \ \delta(i-m)\ di \ = \ P(m)$