Derivation is a linear operation, so $ \frac{\partial G}{\partial x_1} = \frac{\partial}{\partial x_1}\int_0^{x_1}g_1(x,0)\,\mathrm dx + \frac{\partial}{\partial x_1}\int_0^{x_2}g_2(x_1,y)\,\mathrm dy $ The first term is the derivative of the integral (with respect to $x_1$), so you should know how to compute it. For the second term, you need to justify this: $ \frac{\partial}{\partial x_1}\int_0^{x_2}g_2(x_1,y)\,\mathrm dy = \int_0^{x_2}\frac{\partial g_2}{\partial x_1}(x_1,y)\,\mathrm dy. $ There should be a theorem somewhere in your notes/books about that. Finally, just use the property $\partial g_2/\partial x_1=\partial g_1/\partial x_2$, and then the fundamental theorem of calculus should seal the deal.