Word Count : 495

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To show that the function \( f(x, y) \) is differentiable at \( (0, 0) \), we need to verify two conditions: 1. Partial derivatives exist at \( (0, 0) \): Compute the partial derivatives \( f_x(0, 0) \) and \( f_y(0, 0) \). 2. The function is well-approximated by its linear approximation at \( (0, 0) \): Show that the limit \[ \lim_{(h, k) \to (0, 0)} \frac{f(h, k) - f(0, 0) - f_x(0, 0)h - f_y(0, 0)k}{\sqrt{h^2 + k^2}} = 0 \] holds. --- ### Step 1: Compute the partial derivatives \( f_x(0, 0) \) and \( f_y(0, 0) \) The partial derivative \( f_x(0, 0) \) is defined as: \[ f_x(0, 0) = \lim_{h \to 0} \frac{f(h, 0) - f(0, ___ ___ _______ _______ ___ ___ __________ ________ ____ ____ _________.
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