Definition 26.6.1 (Partial Derivative).label Let $E_{1}, E_{2}$ be TVSs over $K \in \RC$, $\sigma_{1} \subset \mathfrak{B}(E_{1})$ and $\sigma_{2} \subset \mathfrak{B}(E_{2})$ be covering ideals, $F$ be a separated TVS over $K$, $U \subset E_{1} \times E_{2}$ be open, and $f: U \to F$. For each $(x_{0}, y_{0}) \in E$, let $f_{x_0}(y) = f(x_{0}, y)$ and $f_{y_0}(x) = f(x, y_{0})$ be the partial maps of $f$. If $f_{x_0}$ is $\tilde \sigma_{1}$-differentiable for each $x_{0}$, and $f_{y_0}$ is $\tilde \sigma_{2}$-differentiable for each $y_{0}$, then
\[D_{1}f: U \to B_{\sigma_1}(E_{1}; F) \quad (x, y) \mapsto D_{\sigma_1}f_{x}(y)\]
and
\[D_{2}f: U \to B_{\sigma_2}(E_{2}; F) \quad (x, y) \mapsto D_{\sigma_2}f_{y}(x)\]
are the partial derivatives of $f$.