My question is essentially whether taking partial derivatives of a smooth family of analytic functions yields again a smooth family of analytic functions.

The precise question is the following:

Let $f: \mathbb{R}^m\oplus \mathbb{R}^n \to \mathbb{R}$ be a smooth function in two variables $x\in \mathbb{R}^m$ and $y\in\mathbb{R}^n$, such that for each fixed x, the function

$$ y \mapsto f(x.y)$$

is globally analytic, i.e. its Taylor expansion around y=0 converges to f(x,y) for all y.

My question is: do the partial derivatives with respect to one of the x-coordinates $\frac{\partial f}{\partial x^i}$ have the same property, i.e. is $\frac{\partial f}{\partial x^i}$ globally analytic in the above sense for each fixed x?

And what about the case when the Taylor expansion around y=0 converges to f(x,y) only if y lies in some open interval of 0?