I would like to understand how to find the first derivative with respect to $C_i$ of $$\mathcal{L}=\left[\int_{i=0}^1 C_i^{(\eta-1)/\eta}di\right]^{\eta/(\eta-1)} +\lambda\left[S-\int_{i=0}^1P_iC_idi\right]$$
The most difficult part here for me is that I can not figure out how I should treat those integrals. Could you please suggest me some readings or explain me (in an explicit way) how to derive the above expression?
Thank you in advance.
Charlie.
We can write the expression as $$ \mathcal{L}=\left[\int_{0}^1 [C(x)]^\zeta\,dx\right]^{1/\zeta} +\lambda\left[S-\int_{0}^1P(x)C(x)dx\right] $$ where $\zeta := (\eta-1)/\eta$. Then, $$ \frac{\partial \mathcal{L}}{\partial C} = \frac{1}{\zeta}\left[\int_{0}^1 C^\zeta\,dx\right]^{\frac{1-\zeta}{\zeta}}\,\frac{\partial}{\partial C}\left[\int_{0}^1 C^\zeta\,dx\right] - \lambda\,\frac{\partial}{\partial C}\left[\int_{0}^1PC\,dx\right] $$ The derivatives above can be interpreted in terms of the calculus of variations. For example, if $$ I[C] = \int_{0}^1 C(x)^\zeta\,dx $$ then $$ \frac{\delta I}{\delta C} = \frac{d}{dC}\left[C^\zeta\right] = \zeta C^{\zeta-1} $$ You will also find the following informative.
