Question

Calculate all second order derivatives of g(x,y)= xsin(x+y)+e^y

Answer 1

$$\dfrac{\partial g}{\partial x}=\sin(x+y)+x\cos(x+y)$$

$$\dfrac{\partial g}{\partial y}=x\cos(x+y)+e^y$$

$$\dfrac{\partial^2 g}{\partial x^2}=2\cos(x+y)-x\sin(x+y)$$

$$\dfrac{\partial ^2g}{\partial x\partial y}=\cos(x+y)-x\sin(x+y)=\dfrac{\partial ^2z}{\partial y\partial x}$$

$$\dfrac{\partial ^2g}{\partial y^2}=-x\sin(x+y)+e^y$$