= Solution
The $j$-th <partial derivative> is the <vector> limit
$$
\partial_jf(x)=\lim_{h\to0}\frac{f(x+he_j)-f(x)}h,
$$
when it exists. The <total derivative> is a <linear map> $Df(x):\mathbb R^n\to\mathbb R^m$ such that
$$
f(x+h)=f(x)+Df(x)h+o(\|h\|).
$$
The <Jacobian matrix> is $J_f(x)=(\partial_jf_i(x))_{i,j}$. When the <total derivative> exists, all partial derivatives exist and $J_f(x)$ is its <matrix> in the standard bases; its $j$-th column is $Df(x)e_j$. Existence of partial derivatives alone does not guarantee total differentiability.
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