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$$\frac{\partial f}{\partial x} = \lim\limits_{h \rightarrow 0} \frac{f(x + h, y) - f(x,y)}{h}$$

and

$$\frac{\partial f}{\partial x} = \lim\limits_{h \rightarrow 0} \frac{f(x_0 + h, y_0) - f(x_0,y_0)}{h}$$

where the latter evaluates the function at the respective point before plugging it into the definition of the limit. For example, the function ##f(x,y) = \begin{cases}

\frac{x^2 y^4}{x^4 + 6y^8}, & \text{if }(x,y) \neq (0,0) \\

0, & \text{if }(x,y) = (0,0)

\end{cases}##

I want to determine if ##\frac{\partial f}{\partial x}## is differentiable at ##(0,0)##.

Using the second limit definition would make showing the existence of ##\frac{\partial f}{\partial x}## considerably easier, since ##y_0## makes the first term in the limit ##0##, and ##f(x_0,y_0)## is defined to be ##0##.

But using the first definition, we have to evaluate:

$$\frac{(x+h)^2 y^4}{(x+h)^4 + 6y^8} - \frac{x^2 y^4}{x^4 + 6y^8}$$

I'm hoping the "real" or at least usable definition is the second one, but which one is the one we're supposed to use in practice to be technically/mathematically correct?