Solve z=xy+e^y/y^2+1.text{Find}partial^2z/partialxpartialy

$$z = x y + \frac { e ^ { y } } { y ^ { 2 } + 1 } . \text { Find } \frac { \partial ^ { 2 } z } { \partial x \partial y }$$

Solve for x

$$\left\{\begin{matrix}x=-\frac{e^{y}-z-zy^{2}}{y\left(y^{2}+1\right)}\text{, }&y\neq 0\\x\in \mathrm{R}\text{, }&z=1\text{ and }y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by y^{2}+1.

$$z\left(y^{2}+1\right)=xy\left(y^{2}+1\right)+e^{y}$$

Use the distributive property to multiply z by y^{2}+1.

$$zy^{2}+z=xy\left(y^{2}+1\right)+e^{y}$$

Use the distributive property to multiply xy by y^{2}+1.

$$zy^{2}+z=xy^{3}+xy+e^{y}$$

Swap sides so that all variable terms are on the left hand side.

$$xy^{3}+xy+e^{y}=zy^{2}+z$$

Subtract e^{y} from both sides.

$$xy^{3}+xy=zy^{2}+z-e^{y}$$

Combine all terms containing x.

$$\left(y^{3}+y\right)x=zy^{2}+z-e^{y}$$

Divide both sides by y^{3}+y.

$$\frac{\left(y^{3}+y\right)x}{y^{3}+y}=\frac{zy^{2}+z-e^{y}}{y^{3}+y}$$

Dividing by y^{3}+y undoes the multiplication by y^{3}+y.

$$x=\frac{zy^{2}+z-e^{y}}{y^{3}+y}$$

Divide zy^{2}+z-e^{y} by y^{3}+y.

$$x=\frac{zy^{2}+z-e^{y}}{y\left(y^{2}+1\right)}$$