Die relevanten Ableitungen sind

${\displaystyle {}{\frac {\partial f}{\partial x}}=e^{x-y^{2}}\,,}$

${\displaystyle {}{\frac {\partial f}{\partial y}}=-2ye^{x-y^{2}}\,,}$

${\displaystyle {}{\frac {\partial ^{2}f}{\partial ^{2}x}}=e^{x-y^{2}}\,,}$

${\displaystyle {}{\frac {\partial ^{2}f}{\partial ^{2}y}}={\left(-2+4y^{2}\right)}e^{x-y^{2}}\,,}$

${\displaystyle {}{\frac {\partial }{\partial y}}{\frac {\partial f}{\partial x}}=-2ye^{x-y^{2}}\,.}$

Somit sind die Werte der relevanten Ableitungen im Punkt ${\displaystyle {}(1,1)}$ gleich

${\displaystyle {}f(1,1)=1\,,}$

${\displaystyle {}{\frac {\partial f}{\partial x}}(1,1)=1\,,}$

${\displaystyle {}{\frac {\partial f}{\partial y}}(1,1)=-2\,,}$

${\displaystyle {}{\frac {\partial ^{2}f}{\partial ^{2}x}}(1,1)=1\,,}$

${\displaystyle {}{\frac {\partial ^{2}f}{\partial ^{2}y}}(1,1)=2\,,}$

${\displaystyle {}{\frac {\partial }{\partial y}}{\frac {\partial f}{\partial x}}(1,1)=-2\,.}$

Daher ist das Taylor-Polynom der Ordnung zwei gleich

${\displaystyle 1+(x-1)-2(y-1)+{\frac {1}{2}}(x-1)^{2}-2(x-1)(y-1)+(y-1)^{2}.}$