Answer to Question #244546 in Calculus for Susan

Question

Find all 4 of the second order partial derivatives for the function f(x,y)=e^-xy +x^2y cos(x) -y^2

Answer 1

$f_x=-ye^{-xy}+2xycosx-x^2ysinx$

$f_y=-xe^{-xy}+x^2cosx-2y$

$f_{xx}=y^2e^{-xy}+2ycosx-2xysinx-2xysinx-x^2ycosx=$

$=y^2e^{-xy}+ycosx(2-x^2)-4xysinx$

$f_{xy}=f_{yx}=-e^{-xy}+xye^{-xy}+2xcosx-x^2sinx=$

$=e^{-xy}(xy-1)+2xcosx-x^2sinx$

$f_{yy}=x^2e^{-xy}-2$