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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) ________ ________ ___ ______ ___ ______ _____ ________.
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