I would like to compute the derivative of the integral of a complicated rational function $W(x_0, \ldots, x_3 ,y_0, \ldots, y_3,\xi,\eta)$, with respect to any of the $x_0, \ldots, x_3 ,y_0, \ldots, y_3$. For example:
$$ \frac{\partial}{\partial x_0} \int_{-1}^1 \int_{-1}^1 W(x_0, \ldots, x_3 ,y_0, \ldots, y_3 \xi,\eta) \; d\eta \;d\xi $$
I have considered moving the derivative inside the integrals like so
$$ \int_{-1}^1 \int_{-1}^1 \frac{\partial W(x_0, \ldots, x_3 ,y_0, \ldots, y_3,\xi,\eta)}{\partial x_0} \; d\eta \;d\xi $$
But this would lead to expression swell because of the complexity of $W(x_0, \ldots, x_3 ,y_0, \ldots, y_3,\xi,\eta)$. Furthermore, doing this precludes automatic differentiation. I am not quite sure how to proceed and wonder if there are techniques for solving such problems.
Update
$$ W = \mu \left[ 0.5 {{\left( \frac{{v_1} \left( 1-{u_{11}}\right) }{{{\left( \left( 1-{v_0}\right) \left( 1-{u_{11}}\right) -{v_1} {u_{10}}\right) }^{2}}}+\frac{\left( 1-{v_0}\right) {u_{10}}}{{{\left( \left( 1-{v_0}\right) \left( 1-{u_{11}}\right) -{v_1} {u_{10}}\right) }^{2}}}\right) }^{2}}+0.25 {{\left( \frac{{{\left( 1-{u_{11}}\right) }^{2}}}{{{\left( \left( 1-{v_0}\right) \left( 1-{u_{11}}\right) -{v_1} {u_{10}}\right) }^{2}}}+\frac{{{{u_{10}}}^{2}}}{{{\left( \left( 1-{v_0}\right) \left( 1-{u_{11}}\right) -{v_1} {u_{10}}\right) }^{2}}}-1\right) }^{2}}+0.25 {{\left( \frac{{{{v_1}}^{2}}}{{{\left( \left( 1-{v_0}\right) \left( 1-{u_{11}}\right) -{v_1} {u_{10}}\right) }^{2}}}+\frac{{{\left( 1-{v_0}\right) }^{2}}}{{{\left( \left( 1-{v_0}\right) \left( 1-{u_{11}}\right) -{v_1} {u_{10}}\right) }^{2}}}-1\right) }^{2}}\right] +0.5 \lambda {{\left[ 0.5 \left( \frac{{{\left( 1-{u_{11}}\right) }^{2}}}{{{\left( \left( 1-{v_0}\right) \left( 1-{u_{11}}\right) -{v_1} {u_{10}}\right) }^{2}}}+\frac{{{{u_{10}}}^{2}}}{{{\left( \left( 1-{v_0}\right) \left( 1-{u_{11}}\right) -{v_1} {u_{10}}\right) }^{2}}}-1\right) +0.5 \left( \frac{{{{v_1}}^{2}}}{{{\left( \left( 1-{v_0}\right) \left( 1-{u_{11}}\right) -{v_1} {u_{10}}\right) }^{2}}}+\frac{{{\left( 1-{v_0}\right) }^{2}}}{{{\left( \left( 1-{v_0}\right) \left( 1-{u_{11}}\right) -{v_1} {u_{10}}\right) }^{2}}}-1\right) \right] }^{2}} $$
where
$$ u_{00} = \frac{\left( \left( \frac{d}{d \xi } {N_3}\right) {v_3}+\left( \frac{d}{d \xi } {N_2}\right) {v_2}+\left( \frac{d}{d \xi } {N_1}\right) {v_1}+\left( \frac{d}{d \xi } {N_0}\right) {v_0}\right) \left( \frac{d}{d \eta } y\right) -\left( \left( \frac{d}{d \eta } {N_3}\right) {v_3}+\left( \frac{d}{d \eta } {N_2}\right) {v_2}+\left( \frac{d}{d \eta } {N_1}\right) {v_1}+\left( \frac{d}{d \eta } {N_0}\right) {v_0}\right) \left( \frac{d}{d \eta } x\right)}{J} $$
$$ u_{10} = \frac{\left( \left( \frac{d}{d \xi } {N_3}\right) {w_3}+\left( \frac{d}{d \xi } {N_2}\right) {w_2}+\left( \frac{d}{d \xi } {N_1}\right) {w_1}+\left( \frac{d}{d \xi } {N_0}\right) {w_0}\right) \left( \frac{d}{d \eta } y\right) -\left( \left( \frac{d}{d \eta } {N_3}\right) {w_3}+\left( \frac{d}{d \eta } {N_2}\right) {w_2}+\left( \frac{d}{d \eta } {N_1}\right) {w_1}+\left( \frac{d}{d \eta } {N_0}\right) {w_0}\right) \left( \frac{d}{d \eta } x\right) }{J} $$
$$ u_{01} = \frac{\left( \left( \frac{d}{d \eta } {N_3}\right) {v_3}+\left( \frac{d}{d \eta } {N_2}\right) {v_2}+\left( \frac{d}{d \eta } {N_1}\right) {v_1}+\left( \frac{d}{d \eta } {N_0}\right) {v_0}\right) \left( \frac{d}{d \xi } x\right) -\left( \left( \frac{d}{d \xi } {N_3}\right) {v_3}+\left( \frac{d}{d \xi } {N_2}\right) {v_2}+\left( \frac{d}{d \xi } {N_1}\right) {v_1}+\left( \frac{d}{d \xi } {N_0}\right) {v_0}\right) \left( \frac{d}{d \xi } y\right) }{J} $$
$$ u_{11} = \frac{\left( \left( \frac{d}{d \eta } {N_3}\right) {w_3}+\left( \frac{d}{d \eta } {N_2}\right) {w_2}+\left( \frac{d}{d \eta } {N_1}\right) {w_1}+\left( \frac{d}{d \eta } {N_0}\right) {w_0}\right) \left( \frac{d}{d \xi } x\right) -\left( \left( \frac{d}{d \xi } {N_3}\right) {w_3}+\left( \frac{d}{d \xi } {N_2}\right) {w_2}+\left( \frac{d}{d \xi } {N_1}\right) {w_1}+\left( \frac{d}{d \xi } {N_0}\right) {w_0}\right) \left( \frac{d}{d \xi } y\right) }{J} $$
and
$$ J = \left( -\frac{\left( \eta +1\right) \, {x_3}}{4}+\frac{\left( \eta +1\right) \, {x_2}}{4}+\frac{\left( 1-\eta \right) \, {x_1}}{4}-\frac{\left( 1-\eta \right) \, {x_0}}{4}\right) \left( \frac{\left( 1-\xi \right) \, {y_3}}{4}+\frac{\left( \xi +1\right) \, {y_2}}{4}-\frac{\left( \xi +1\right) \, {y_1}}{4}-\frac{\left( 1-\xi \right) \, {y_0}}{4}\right) -\left( \frac{\left( 1-\xi \right) \, {x_3}}{4}+\frac{\left( \xi +1\right) \, {x_2}}{4}-\frac{\left( \xi +1\right) \, {x_1}}{4}-\frac{\left( 1-\xi \right) \, {x_0}}{4}\right) \left( -\frac{\left( \eta +1\right) \, {y_3}}{4}+\frac{\left( \eta +1\right) \, {y_2}}{4}+\frac{\left( 1-\eta \right) \, {y_1}}{4}-\frac{\left( 1-\eta \right) \, {y_0}}{4}\right) $$
$$ N_0 = \frac{\left( 1-\eta \right) \, \left( 1-\xi \right) }{4} $$
$$ N_1 = \frac{\left( 1-\eta \right) \, \left( \xi +1\right) }{4} $$
$$ N_2 = \frac{\left( \eta +1\right) \, \left( \xi +1\right) }{4} $$
$$ N_3 = \frac{\left( \eta +1\right) \, \left( 1-\xi \right) }{4} $$