Answer
False.
Work Step by Step
According to theorem 13.4.1, a function $f(x, y)$ is said to be differentiable at $\left(x_{0}, y_{0}\right)$ if both the partial derivatives exist and:
\[
\lim _{(\Delta x, \Delta y) \rightarrow(0,0)} \frac{\Delta f-f_{x}\left(x_{0}, y_{0}\right) \Delta x-f_{y}\left(x_{0}, y_{0}\right) \Delta y}{(\sqrt{\Delta y})^{2}+(\Delta x)^{2}}=0
\]
Therefore, in the above problem, we are told about only the partial derivatives; no information is given about the limit. Thus, the statement is False.
