Show a function defined by an integral is differentiable [closed]

Let $f=f(x,y)$ be given by: $$f(x,y)=\int_x^{x+y}\int_{x-y}^ye^{s^2+t^2}dsdt$$
Show that $f$ is differentiable and calculate $\nabla f(x,y)$


Usually I would write my attempt, however, I have no idea what to do


Edit: Apparently you can break this double integral into a product of integrals
$\int_x^{x+y}\int_{x-y}^ye^{s^2+t^2}dsdt=\int_x^{x+y}\int_{x-y}^ye^{s^2}e^{t^2}dsdt=\int_x^{x+y}e^{t^2}(\int_{x-y}^ye^{s^2}ds)dt=\int_x^{x+y}e^{t^2}dt\int_{x-y}^ye^{s^2}ds$
Then, we use the Fundamental Theorem of Calculus to find the partial derivatives:
$$\frac{\partial}{\partial x}f(x,y)=(e^{(x+y)^2}-e^{x^2})\int_{x-y}^y e^{s^2}ds+\int_x^{x+y}e^{t^2}dt\cdot(-e^{(x-y)^2})$$
$$\frac{\partial}{\partial y}f(x,y)=e^{(x+y)^2}\int_{x-y}^ye^{s^2}ds+\int_x^{x+y}e^{t^2}dt\cdot(e^{y^2}-e^{(x-y)^2})$$
I believe I can't simplify this any further
On the question of wether $f$ is differentiable, I still don't know what to do. The FTC shows that, fixing one of the variables, then $f$ is differentiable as a function of the other variable, but all this shows is that $f$ has partial derivatives, not that it is differentiable