## Calculus 8th Edition

$f_{xyy}=f_{yxy}=f_{yyx}$
Consider, we have $f_{xyy}=(f_{xy})_{y}$ ...(1) $f_{yxy}=(f_{yx})_{y}$ ...(2) From the equations (1) and (2), we get $(f_{xy})_{y}=(f_{xy})_{y}=f_{xyy}$ This yields: $f_{yxy}=f_{xyy}$ Since, $f_{yyx}=(f_{y})_{yx}\\(f_{y})_{yx}$=$(f_{y})_{xy}$ Thus, we have $f_{yyx}=(f_{y})_{xy}=f_{yxy}$ $\implies f_{yxy}=f_{yxy},$ Transitive property of equality defines as: $f_{xyy}=f_{yxy}=f_{yyx}$