Could you please help me with understanding how to take the partial derivative of

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

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.