Word Count : 450

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To determine whether the function \( f(x, y) = \frac{x^2 - y^2}{x^2 + y^2} \) satisfies Schwarz's theorem at the point \((1, 1)\), we need to check if the mixed partial derivatives are equal at this point. Schwarz's theorem states that if the second partial derivatives of a function are continuous in a neighborhood of a point, then the mixed partial derivatives at that point should be equal. Specifically, we need to check if: \[ f_{xy}(1, 1) = f_{yx}(1, 1) \] Let’s compute these partial derivatives. 1. Compute \( f_x(x, y) \) (the partial derivative of \( f \) with respect to \( x \)): \[ f(x, y) = \frac{x^2 - y^2}{x^2 ______ _____ __________ _____ _______ ___ ____ ___ __________ ___.
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