Question

Compute fxy and fyx for the function f(x,y) = e^(x+y) +9x^2 +2xy at (1,2).

Answer 1

$f(x,y)=\exp^{x+y}+9x^2+2xy.$

Find $f_{xy}$ and $f_{yx}$ at (1,2).

Find first $f_x$ and $f_y$

$f_x=\frac{\partial}{\partial x}(\exp^{x+y}+9x^2+2xy)=\exp^{x+y}+18x+2y,$

$f_y=\frac{\partial}{\partial y}(\exp^{x+y}+9x^2+2xy)=\exp^{x+y}+2x.$

To find $f_{xy}$ and $f_{yx}$ take the partial derivative of $f_{x}$ with respect to $y$ and $f_{y}$ with respect to $x$ respectively.

$f_{xy}=\frac{\partial}{\partial y}(\exp^{x+y}+18x+2y)=\exp^{x+y}+2,$

$f_{yx}=\frac{\partial}{\partial x}(\exp^{x+y}+2x)=\exp^{x+y}+2.$

So,

$f_{xy}(1,2)=f_{yx}(1,2)=\exp^{1+2}+2=\exp^{3}+2$.