Let $U \subset \mathbb{R}^m$ be open, let $a \in U$, and let $f: U \to \mathbb{R}^n$ be differentiable at $a$ with total derivative $Df_a: \mathbb{R}^m \to \mathbb{R}^n$. Let $(e_1,\ldots,e_m)$ denote the standard basis of $\mathbb{R}^m$. Then, for every $i \in \{1,\ldots,m\}$, the [partial derivative](/page/Partial%20Derivative) $\partial_{x_i}f(a)$ exists as a vector in $\mathbb{R}^n$, and

\begin{align*} \partial_{x_i}f(a)=Df_a(e_i). \end{align*}

Equivalently, if $f=(f_1,\ldots,f_n)$, then every scalar partial derivative $\partial_{x_i}f_j(a)$ exists and the Jacobian matrix $Jf_a \in \mathbb{R}^{n \times m}$ represents $Df_a$ in the standard bases, with entries

\begin{align*} (Jf_a)_{ji}=\partial_{x_i}f_j(a). \end{align*}