Problem on differentiation of multivariable function

Question

The question is to find the value of $f$ differentiated partially with respect to $x$ (say $f_x$) at $(0,0)$ And the find the value of $f$ differentiated partially with respect to $y$ (say $f_y$) at $(0,0$)

I find $f_x(0,0) =1$ But I am unable to find $f_y(0,0)$

Please help NOTE: Consider $(x,y)\in \mathbb R^2$ in the image.

Answer 1

It is just 0. Recall partial derivative definition:

In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant

So fy(0, 0) is just going to be:

$$fy(0, 0) = \lim_{\Delta y \to 0} \frac{f(0, \Delta y) - f(0, 0)}{\Delta y}$$

And from the function definition, when $x = 0, f = 0$, so $f(0, \Delta y) - f(0, 0)$ is always zero, which makes the above limit to be zero. And more generally, for this function $fy(0, y) = 0, \forall y$.