We can prove that $f(x,y)$ is not differentiable at $(0,0)$ from the definition of differentiability for multivariate functions: $$ \triangle f-f(x_0,y_0)-f_x\cdot\triangle x-f_y\cdot\triangle y=\epsilon_1\cdot \triangle x+\epsilon_2\cdot\triangle y $$ by finding that epsilons that don't go to $0$.
I guess geometrically we can see that there's a "sharp" downward decline at $(0,0)$. But what is the geometrical explanation for the existence of partial derivatives?
Because if we slice the function graph over $x$ or $y$ we still see the sharp downward decline.
Here's a link to the interactive graph: https://ggbm.at/ukVg8PaR