Answer to Question #108254 in Calculus for TUHIN SUBHRA DAS

Question

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

Answer 1

Find first $f_x$ and $f_y$

$f_x=\frac{\partial}{\partial x}(e^{x}+y\sin x+9x^2+2xy)=e^x+y\cos x+18x+2y,$ $f_y=\frac{\partial}{\partial y}(e^{x}+y\sin x+9x^2+2xy)=\sin x+2x.$

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.

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

$f_{yx}=\frac{\partial}{\partial x}(\sin x+2x)=\cos x+2.$

Then,

$f_{xy}(1,2)=f_{yx}(1,2)=\cos 1+2.$