In Exercises 21 through 23, find f12(0,0) and f21(0,0), if...

In Exercises 21 through 23, find $f_{12}(0,0)$ and $f_{21}(0,0)$, if they exist.
$$
f(x, y)= \begin{cases}\frac{2 x y}{x^{2}+y^{2}} & \text { if }(x, y) \neq(0,0) \\ 0 & \text { if }(x, y)=(0,0)\end{cases}
$$

Key Concepts

Path Dependency in Multivariable Limits

Path dependency concerns the situation where the value of a limit, and hence a derivative, depends on the direction or path along which the point is approached. This concept is critical when assessing the existence of derivatives at points where the function's definition changes, as different approaches can yield different derivative values.

Limit Definition of Derivatives

The limit definition is used to rigorously compute derivatives, particularly at points where the function might be defined piecewise or exhibits problematic behavior. It involves taking the limit of the difference quotient as the increment approaches zero, ensuring a precise understanding of the derivative at a given point.

Partial Derivatives

Partial derivatives measure the rate at which a multivariable function changes with respect to one variable while keeping the other variables constant. They are foundational to the study of functions of several variables and are essential in analyzing behavior around specific points, particularly when using a piecewise definition.

Mixed Partial Derivatives

Mixed partial derivatives are second order derivatives where the differentiation is performed with respect to different variables in succession. They provide insight into how the rate of change in one variable is affected by changes in another, and their existence and equality (under certain conditions) can indicate smoothness and consistency in the function's curvature.

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