TITLE: Study differentiability of a multi variable function
QUESTION [0 upvotes]: Let $f(x,y)=\sqrt{x^2+(y-1)^2}$. Study the differentiability of the function at the point $(0,1).$

I know that the derivative of a multi variable function is calculated as follows:
$$\lim_{h\to0}\frac {\|f(x+h,y+h)-f(x,y)-J(h)\|_{\mathbb R}}{\|h\|_{\mathbb{R}^2}}$$
How do I actually use this on this function?

REPLY [2 votes]: Check the partial derivatives at $\;(0,1)\;$:
$$\begin{cases}f'_x=\cfrac x{\sqrt{x^2+(y-1)^2}}\implies f'_x(0,1)=...\text{doesn't exist}\\{}\\
f'(y)=\frac {y-1}{\sqrt{x^2+(y-1)^2}}\implies f'_y(0,1)=...\text{ doesn't exist}\end{cases}$$
and thus $\;f\;$ cannot be differentiable at $\;(0,1)\;$
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