Assume that $f = f(x,y)$ is a function defined on $E × (a,b).$ For each fixed $y ∈ (a,b),$ $f$ is integrable with respect to $x$ on $E$, and for each fixed $x ∈ E$, $f$ is differentiable with respect to $y$ on $(a, b).$ Assume that there is a function $g ∈ L_1(E),$ such that $| \frac{d}{dy} f(x,y)| ≤ g(x),$ $(x,y) ∈ E × (a,b).$ Show that $\int_E f (x, y) dx$ is differentiable with respect to $y$ and $\frac{d}{dy}\int_E f(x,y)dx=\int_E \frac{d}{dy}f(x,y)dx.$
I considered using the functions $f_n(x) = n[f(x, y + n^{-1}) - f(x, y)]$. The $f_n$ are integrable and tend to $\frac{d}{dy}f(x, y)$ which is integrable since $\leq g(x)$. However, all this gives me to use is Fatou since I can't dominate the $f_n$ by any functions. Any help would be awesome! Thanks!