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Understanding partial derivatives begins with visualizing a surface representing a function of two variables, such as \( f(x, y) \). A partial derivative is like a snapshot that captures the rate of change of the function with respect to one variable while keeping the other constant.

Picture yourself hiking on a mountain range represented by a function; taking a step forward but not sideways measures the rate of change in your elevation with respect to your forward movement. That's what a partial derivative does. Formally, for a function \( f(x, y) \), the partial derivative with respect to \( x \), denoted as \(\frac{\partial f}{\partial x}\), is found by differentiating \( f \) with respect to \( x \) while treating \( y \) as a constant. Similarly, the partial derivative with respect to \( y \), \(\frac{\partial f}{\partial y}\), is found by differentiating with respect to \( y \) while \( x \) is held constant.

In the given problem, both the partial derivatives at the origin (0,0) are computed, resulting in 0 for both cases. This indicates that the function's surface is flat along the \( x \) and \( y \) axes at that point. However, this information alone doesn't give a complete picture of the surface's behavior at (0,0).

The traditional definition of a derivative of a function at a certain point is rooted in the concept of limits. For a single variable function, the derivative at a point \( a \) is defined as the limit of the difference quotient as \( h \) approaches 0: \[\frac{f(a+h) - f(a)}{h}.\] This formula tells us the slope of the tangent line to the function at \( a \) and signifies the instantaneous rate of change.

In the context of multiple variables, similar principles apply, and the limit is used to define how a function changes in the direction of each variable separately. The function's tendency towards a particular slope or direction when approached from various paths at a given point captures the essence of a derivative in multiple dimensions.

In our exercise, the limit definition is used to illustrate that while the existence of the partial derivatives (slopes in the direction of \( x \)) and \( y \) at (0,0) are confirmed, they do not necessarily prove the differentiability of the function at that point.

A function is considered to be differentiable at a point if it can be locally approximated by a plane at that point. The hallmark of differentiability is that as you zoom in on the graph of the function at a differentiable point, it begins to resemble a flat plane—the tangent plane. There are formal criteria to establish this.

One of the central criteria is that the function must have partial derivatives at the point in question and, more importantly, that the function increment can be expressed using these derivatives and a higher-order residual that diminishes as it nears the point.

In our problem, even though the partial derivatives exist at (0,0), the function fails to meet the requisite conditions of differentiability because it does not approach a linear function as we zoom in. The limit involving the function and its linear approximation (said to be 0 in this context) does not approach 0 (but rather \( \sqrt{2} \) in the exercise), signifying that the function's behavior is too erratic at (0,0) to be smoothed out by a tangent plane.