Answer
$$g_{xx}=e^{x}-y \ \sin(x); \\ g_{yy}=0 ; \\g_{xy}=g_{yx}=cos(x)$$
Work Step by Step
$$g(x,y)=e^{x}+y \ \sin(x); \\g_{x}=e^{x}+y \ \cos(x); \\g_{xx}=e^{x}-y \ \sin(x); \\g_{yx}=\cos(x) \\g_{y}=\sin(x); \\g_{yy}=0; \\g_{xy}=g_{yx}$$
The second partial derivatives are as follows: $$g_{xx}=e^{x}-y \ \sin(x); \\ g_{yy}=0 ; \\g_{xy}=g_{yx}=cos(x)$$
