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The problem you shared asks to use the definition of partial derivatives as limits to find the partial derivatives \( f_x(0, 0) \) and \( f_y(0, 0) \) for the given function \( f(x, y) \), which is defined piecewise:

\[
f(x, y) =
\begin{cases}
\frac{-xy(x^2 - y^2)}{x^2 + y^2} & \text{if } (x, y) \neq (0, 0) \\
0 & \text{if } (x, y) = (0, 0)
\end{cases}
\]

### Step 1: Definition of partial derivatives
The partial derivative of \( f \) with respect to \( x \) at \( (0, 0) \) is given by the limit definition:

\[
f_x(0, 0) = \lim_{h \to 0} \frac{f(h, 0) - f(0, 0)}{h}
\]

Similarly, for the partial derivative with respect to \( y \):

\[
f_y(0, 0) = \lim_{h \to 0} \frac{f(0, h) - f(0, 0)}{h}
\]

### Step 2: Computing \( f_x(0, 0) \)

To compute \( f_x(0, 0) \

Use the definition of partial derivatives as limits to find the indicated partial derivatives f_x(0, 0) and f_y(0, 0) for the given function f(x, y).

Learn how to compute the partial derivatives f_x(0, 0) and f_y(0, 0) using the limit definition for a piecewise function in multivariable calculus. This problem involves key concepts in limits and partial derivatives, ideal for college-level students studying advanced calculus.

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