Vtyjkyuk

The problem:
Let $ X_1,X_2,cdots ,X_n $ be a random sample from the uniform distribution with pdf $f(x;theta _1,theta _2)=frac12theta _2 , theta _1-theta _2<x<theta _1+theta , $ where$ -infty <theta _1<infty ,0<theta _2<infty $ and the pdf is equal to zero elsewhere.

Show that $Y_1=min(X_i) $ and $Y_n=max(X_i)$, the joint sufficient statistic for $theta _1 $ and $theta _2$ are complete.

My question:
After some calculations, I have found that showing the completeness is equivalent to showing the statement below.

$int_theta _1-theta _2^theta _1+theta _2 int_theta _1-theta_2^y_n u(y_1,y_n)(y_n-y_1)^n-2/(2theta _2)^ndy_1dy_n=0 $ for all $-infty <theta _1<infty ,0<theta _2<infty$ $Rightarrow u(y_1,y_n)=0$

I looked at the solution and it was written like this: Multiply by $(2theta _n)^n$ and differentiate first with respect to $theta_1$ and then $theta_2$. This finally yieds $u(theta _1-theta _2,theta _1+theta _2)=0 $ , for all $(theta _1,theta _2)$, which implies that $u(y_1,y_n)=0$.

I'm stuck in differentiating the double integrals. I have seen differentiaing the single integral using the fundamental theorem of calculus, but in this case , I don't know how to differentiate.