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The problem on the board involves calculating the total differential \( dz \) of the function \( z = \frac{x}{y} \), given the initial values of \( x \), \( y \), and the changes \( \Delta x \) and \( \Delta y \).

### Steps to solve for \( dz \):
1. **Formula for total differential**:
   The total differential \( dz \) of a function \( z(x, y) \) is given by:
   \[
   dz = \frac{\partial z}{\partial x} dx + \frac{\partial z}{\partial y} dy
   \]
   
2. **Partial derivatives**:
   For \( z = \frac{x}{y} \):
   - The partial derivative with respect to \( x \):
     \[
     \frac{\partial z}{\partial x} = \frac{1}{y}
     \]
   - The partial derivative with respect to \( y \):
     \[
     \frac{\partial z}{\partial y} = -\frac{x}{y^2}
     \]
   
3. **Substitute into the total differential formula**:
   \[
   dz = \frac{1}{y} dx - \frac{x}{y^2} dy
   \]
   
4. **Substitute the given values**:
   Given:
   \[
   x = 10, \quad y = 20, \quad \Delta x = 1, \quad \Delta y = 2
   \]
   This means \( dx = 1 \) and \( dy = 2 \).
   Substitute these into the formula for \( dz \):
   \[
   dz = \frac{1}{20} \times 1 - \frac{10}{20^2} \times 2
   \]
   Simplifying:
   \[
   dz = \frac{1}{20} - \frac{10}{400} \times 2 = \frac{1}{20} - \frac{20}{400}
   \]
   \[
   dz = \frac{1}{20} - \frac{1}{20} = 0
   \]

### Conclusion:
The total differential \( dz \) is 0.

Would you like more details, or have any questions about this solution?

### Related questions:
1. What is the geometric interpretation of the total differential?
2. How does the differential relate to the approximation of small changes in multivariable functions?
3. What would the result be if \( \Delta y \) were zero?
4. How does the total derivative change if the function were \( z = x^2 + y^2 \)?
5. How does the chain rule apply to partial derivatives?

### Tip:
For small changes in variables, the total differential gives an excellent approximation for the actual change in the function, which is why it's commonly used in calculus and physics.

Calculate the total differential dz of the function z = x/y given x = 10, y = 20, Δx = 1, and Δy = 2.

Learn how to calculate the total differential dz for the function z = x/y with specific values for x, y, Δx, and Δy. This problem involves the concepts of multivariable calculus, including partial derivatives and total differentials. Suitable for advanced high school and college students.

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