I need to proof that this function is differentiable in all the points of its domain. I know that this is true if the function is a function $\in C^k$ and a function is $C^k$ if is composition of $C^k$ functions. However considering
$$ f(x,y)=x^3+e^{x+y^2} $$
Surely I can say that it is a composition of $C^k$ functions. Moreover I can partially derivate the function and proof that the derivates are composition of continuos functions. There are some other method maybe with some calculation which I can use?
Some aid? Thanks
There is a result stating that a function $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ is differentiable at a point $(x,y)$ if its partial derivatives exist in some neighbourhood of $(x,y)$ and are continuous at $(x,y)$ - e.g. Theorem 2.3.4 of Multidimensional Real Analysis I by Kolk & Duistermaat. So you need only calculate the partial derivatives of $f$ and observe that they exist and are continuous everywhere - this shows that $f$ is differentiable everywhere.