How do I show that a function $f:\mathbb{R}^2\to \mathbb{R}$ is differentiable at every point $(x,y)\in\mathbb{R}^2$?
For example, $$f(x,y)=\ln(1+x^4+y^2)$$
Is it enough to prove that the resulting function in $\mathbb{R}$ is continuous and exists for all $(x,y) $ or is there something more I have to do?
For the partial derivatives we have
$f_x(x,y)=\frac{4x^3}{1+x^4+y^2}$ and $f_y(x,y)=\frac{2y}{1+x^4+y^2}$.
These partial derivatives are continuous on $ \mathbb R^2$. Hence, $f$ is differentiable on $ \mathbb R^2$.