Equivalent condition for differentiability on partial derivatives
I guess you are looking for the below theorem: Reference Mathematical Analysis, by Tom Apostol page $357$.
$\textbf{Theorem.}$ Assume that one of the partial derivatives $D_{1}\mathbf{f},\cdots D_{n}\mathbf{f}$ exists at $\mathbf{c}$ and that the remaining $n-1$ partial derivatives exists in some $n$-ball $B(\mathbf{c})$, and are continuous at $\mathbf{c}$. Then $f$ is differentiable at $\mathbf{c}$.