# Derivative of double integral

1. Feb 23, 2013

### arfie

Hi everyone!

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

Suppose:

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

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

$L= \sum_{i = 1}^{J} \Big[\int_S \omega (y,\alpha) V^e(p_i,y+r(y,\alpha)-T_i,g_i,\alpha)$ $a_i(y,\alpha)f(y,\alpha)\,dy\,d\alpha + \omega_R(R/J+\int_0^{p_i/(1+t_i)}H^i_s(z)\,dz)\Big]$

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

How can I solve for FOC for the lagrangian function with respect to $r(y,\alpha)$? 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!

2. Feb 25, 2013

### joeblow

Definitely not enough info here for anyone to understand what's going on. For instance, "solve for FOC for the lagrangian function with respect to r(y,a)" is meaningless to me.

3. Feb 25, 2013

### arfie

I mean "how to get first derivative of L w.r.t r(y,a) which is dL/dr(y,a)?"

4. Feb 25, 2013

### micromass

Staff Emeritus
Sorry, but you're going to have to give way more information. Start by defining all the notation you use. Describe the actual problem that's given. Maybe give a reference to where you have seen this.

5. Feb 25, 2013

### arfie

Sorry if it's not that clear. The reference is here: http://www.nber.org/papers/w17251.pdf
My problem is in page 14, equation (20). I don't know how to get it.