Word Count : 351

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We are to compute $f_{xy}(0,0)$ and $f_{yx}(0,0)$ using first principles (definitions) for the function: $$ f(x,y)= \begin{cases} \dfrac{x^2y}{\sqrt{x^2+y^2}}, & (x,y)\neq (0,0)\\[4pt] 0, & (x,y)=(0,0) \end{cases} $$ --- ### Step 1: Recall definitions The mixed partial derivatives at $(0,0)$ are defined as: $$ f_{xy}(0,0) = \lim_{h\to 0} \frac{ f_x(0,h) - f_x(0,0)}{h} $$ $$ f_{yx}(0,0) = \lim_{h\to 0} \frac{ f_y(h,0) - f_y(0,0)}{h} $$ So we must first find $f_x(x,y)$ and $f_y(x,y)$ at and near $(0,0)$. --- ### Step 2: Compute $f_x(0,y)$ From the definition of partial derivative: $$ f_x(0,y) __________ _________ _______ ___ __________ __________ ___ ____.
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