A function $f: \R^n \to \R^m$ is differentiable at $p \in \R^n$ if $f$ can be approximable by a constant plus a linear term plus a remainder.
If approximation exists, then the differential is unique (and has a formula)
If the partial derivative exists, and is continuous, then the total derivative exists. (Proof: show the remainder is sublinear, by examine componentwise)
(key) total derivative satisfies all the nice properties.
A function is differentiable at a point $p$, iff all its components are differentiable.
Two mean value theorems: a crude one on length; a more precise one, using averaging.
Differentiation commute with integration.