= Solution
Writing $f=(f_1,\ldots,f_m)$, the $j$th <partial derivative> at $x$ is the vector limit
$$
\partial_j f(x)=\lim_{t\to0}\frac{f(x+te_j)-f(x)}t,
$$
when it exists. The <total derivative> is a <linear map> $Df(x):\mathbb R^n\to\mathbb R^m$ satisfying
$$
f(x+h)=f(x)+Df(x)h+o(\|h\|).
$$
The <Jacobian matrix> has entries $J_f(x)_{ij}=\partial_j f_i(x)$. If the <total derivative> exists, its matrix in the standard <basis>[bases] is precisely $J_f(x)$, because substitution of $h=te_j$ identifies every column. Existence of the <partial derivative>[partial derivatives] alone does not in general imply existence of the <total derivative>.
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