piecewise multivariate partial derivative

Problem

Given a piecewise function $f$: $f(x,y)=\{\begin{array}{ll}g(x)& (x,y)\ne (0,0)\\ h(x)& (x,y)=(0,0)\end{array}$ Find its partial derivatives at every point $(x,y)$

For any point $(x,y)\ne (0,0)$,

$$\begin{array}{rl}\frac{\partial f}{\partial x}(x,y)& =\frac{\partial g}{\partial x}(x,y)\\ \frac{\partial f}{\partial y}(x,y)& =\frac{\partial g}{\partial y}(x,y)\end{array}$$

Plug the $y=0$ into $\frac{\partial f}{\partial x}$, and evaluate. We need to check if the second derivative exists at this point. If not, then the partial derivative doesn’t exist at $(0,0)$. Do the same for $\frac{\partial f}{\partial y}$.

If the second partial derivative does exist at $(0,0)$ (piecewise is OK), and the left and right side first derivatives evaluate to the same value at $(0,0)$, then the partial derivative exist at $(0,0)$.