I am eager to learn more about the differential. I was told that it is a good practice to compute the differential independent of a choice of a basis using the following definition
Definition: $f: \Omega \to \mathbb{R}^m$ with $\Omega \subset \mathbb{R}^n $ is said to be differentiable at $x_0 \in \Omega$ if there exists a linear mapping $A: \mathbb{R}^n \to \mathbb{R}^m$ such that $$\lim_{x \to x_0} \frac{f(x)-f(x_0)-A(x-x_0)}{||x-x_0||}=0 $$
I am aware that it is easy to show that a function is differentiable using the Jacobi Matrix, but the Jacobi Matrix makes a choice of a basis, using the Standardbasis in $\mathbb{R}^n$. I am wondering how I could find the differential for example the following exercise: $$f: \mathbb{R}^2 \to \mathbb{R}, \ f(x,y)=xy^2$$ Unfortunately I cannot link any work here yet because I don't know where to start, my only method so far is computing the Jacobi-Matrix and then discuss its properties (like existence of every partial derivative, continuity of the entries)