Study differentiability of a multi variable function

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?

• That's not the definition. What you wrote doesn't make sense. Is $h$ a real number or an element of $\mathbb{R}^2$? – José Carlos Santos Jan 21 at 18:29
• It's on the wikipedia page ......... en.wikipedia.org/wiki/… – C. Cristi Jan 21 at 18:30
• My guess is, it's an element of $\mathbb R^2$ – C. Cristi Jan 21 at 18:31
• If it's an element of $\mathbb{R}^2$, then what's the meaning of $x+h$? – José Carlos Santos Jan 21 at 18:34
• @JoséCarlosSantos You're right, I know, but I have no idea, I just want to study the differentiability and that's the definition I found on that site – C. Cristi Jan 21 at 18:35

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 $$\;(0,1)\;$$