Understanding derivatives

Instead of going into the formal notion of continuity, differentiability and so on let us become a physicist for a while and assume all functions are both continuous and differentiable.

We are more or less concerned with (in the general case) functions $f:{\mathbb{R}}^m⟶{\mathbb{R}}^n$.

But first, let us look at the simple $f:\mathbb{R}\to \mathbb{R}$
we know that the derivative at $x_0$ is the slope of the line tangent to the graph $(x,f(x))$ at $x_0$.
Let us try to write the equation for the tangent line itself. $$ T \equiv f'(x_{0})(x-x_{0}) =y - f(x_{0})$$ Equivalently, the line $T$ is given by the equation $$y = f'(x_{0})x + (f(x_{0}) - x_{0}f'(x_{0}))$$
This line $T$ is at least an affine subspace of ${\mathbb{R}}^n$. So this line supports the graph as a tangent at $x_0$. for any $x$, what is the distance between the ouput of this line and the function $f$? well, it is simply $y-f(x)$ =