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I really need help for this. I have to read a paper in economics where some parts I don't understand.

Suppose:

[itex]S \equiv [\alpha,\bar{\alpha}]x[y,\bar{y}] [/itex]

[itex]V^e(p_j,g_j,y+r(r\alpha)-T_j,\alpha)\equiv \underset{h}{\text{max}}U(y+r(r,\alpha)-T_j,p_jh,h,g_j;\alpha)[/itex]

And then from maximization problem I have a Lagrangian function as follows:

[itex]L=

\sum_{i = 1}^{J} \Big[\int_S \omega (y,\alpha) V^e(p_i,y+r(y,\alpha)-T_i,g_i,\alpha)[/itex] [itex]a_i(y,\alpha)f(y,\alpha)\,dy\,d\alpha

+ \omega_R(R/J+\int_0^{p_i/(1+t_i)}H^i_s(z)\,dz)\Big][/itex]

[itex]\lambda_1[R+\int_S r(y,\alpha)f(y,\alpha)\,dy\,d\alpha]+\lambda_2[\int_S h_d(p_i,y+r(y,\alpha)-T_i,g_i,\alpha)a_i(y,\alpha)f(y,\alpha)\,dy\,d \alpha-H_S^i][/itex]

How can I solve for FOC for the lagrangian function with respect to [itex]r(y,\alpha)[/itex]? What rules should I use? I guess it has something to do with Leibniz's rule and chain rule but I'm not sure. Thanks!

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# Derivative of double integral

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