Solution
= Solution
The <total derivative> is a <linear map> $Df(a):\mathbb R^n\to\mathbb R^n$ such that
$$
\boxed{f(a+h)=f(a)+Df(a)h+r(h),\qquad\frac{\|r(h)\|}{\|h\|}\longrightarrow0\quad(h\to0).}
$$
It is unique: the difference of two such <linear maps>, evaluated on $h=tv$, must vanish on every fixed <vector> $v$ after division by $t$ and passage to the limit.