Question:
find \(f_{x}, f_{y}, f_{xx}, f_{yy}, f_{xy},and f_{yx} for:\)
\[f\left(x,y\right) = {x}^{2}y+3{x}^{2}+4y-5\]
Answer:
\[f_{x}\left(x,y\right) = 2xy + 6x\]
\[f_{y}\left(x,y\right) = {x}^{2}+4\]
\[f_{xx}\left(x,y\right) = \left( 2xy + 6x \right)dx = 2y + 6\]
\[f_{yy}\left( x,y\right) = \left({x}^{2}+4 \right)dy = 0\]
\[f_{xy}\left(x,y\right) = \left( 2xy + 6x \right) dy = 2x\]
\[f_{yx}\left(x,y\right) = \left( {x}^{2}+4 \right)dx = 2x\]