To solve the given problem of finding the total differential for each function, we will need to differentiate each function with respect to its variables. The total differential $dz$ of a function $z = f(x, y)$ is given by:

$dz = \frac{\partial z}{\partial x} dx + \frac{\partial z}{\partial y} dy$

Let's find the total differentials for each function provided in the image.

1. $z = xy + \frac{x}{y}$

   To find $dz$:

   $\frac{\partial z}{\partial x} = y + \frac{1}{y}$ $\frac{\partial z}{\partial y} = x - \frac{x}{y^2}$

   Therefore, the total differential is:

   $dz = \left( y + \frac{1}{y} \right) dx + \left( x - \frac{x}{y^2} \right) dy$

2. $z = e^{\frac{y}{x}}$

   To find $dz$:

   Let $u = \frac{y}{x}$, then $z = e^u$.

   $\frac{\partial z}{\partial x} = \frac{\partial z}{\partial u} \cdot \frac{\partial u}{\partial x} = e^u \cdot \left( -\frac{y}{x^2} \right) = -\frac{y}{x^2} e^{\frac{y}{x}}$

   $\frac{\partial z}{\partial y} = \frac{\partial z}{\partial u} \cdot \frac{\partial u}{\partial y} = e^u \cdot \frac{1}{x} = \frac{1}{x} e^{\frac{y}{x}}$

   Therefore, the total differential is:

   $dz = -\frac{y}{x^2} e^{\frac{y}{x}} dx + \frac{1}{x} e^{\frac{y}{x}} dy$

3. $z = \frac{y}{\sqrt{x^2 + y^2}}$

   To find $dz$:

   Let $u = x^2 + y^2$, then $z = \frac{y}{\sqrt{u}}$.

   $\frac{\partial z}{\partial x} = \frac{\partial z}{\partial u} \cdot \frac{\partial u}{\partial x} = \left( -\frac{1}{2} \cdot \frac{y}{u^{3/2}} \right) \cdot 2x = -\frac{xy}{(x^2 + y^2)^{3/2}}$

   $\frac{\partial z}{\partial y} = \frac{\partial z}{\partial u} \cdot \frac{\partial u}{\partial y} = \left( -\frac{1}{2} \cdot \frac{y}{u^{3/2}} \right) \cdot 2y + \frac{1}{\sqrt{u}} = -\frac{y^2}{(x^2 + y^2)^{3/2}} + \frac{1}{\sqrt{x^2 + y^2}}$

   Simplifying the second term:

   $\frac{\partial z}{\partial y} = -\frac{y^2}{(x^2 + y^2)^{3/2}} + \frac{x^2 + y^2}{(x^2 + y^2)^{3/2}} = \frac{x^2}{(x^2 + y^2)^{3/2}}$

   Therefore, the total differential is:

   $dz = -\frac{xy}{(x^2 + y^2)^{3/2}} dx + \frac{x^2}{(x^2 + y^2)^{3/2}} dy$

Would you like any more details on any of these differentials or have any questions? Here are five related questions you might consider:

1. How do you find the partial derivative of a function?
2. What is the geometric interpretation of the total differential?
3. How do you apply the chain rule in multivariable calculus?
4. Can you explain the significance of the total differential in physics?
5. How do you handle differentials when working with implicit functions?

Tip: When calculating partial derivatives, ensure you treat all other variables as constants and only differentiate with respect to the variable of interest.