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The question in the image asks to find the differential \( dz \) of the function \( z = \sqrt{xyy} \), evaluated at the point \( M(1, 1) \).

### Step-by-Step Solution:

1. **Function Simplification**:
   \[
   z = \sqrt{xyy} = \sqrt{x y^2} = y \sqrt{x}
   \]
   
2. **Partial Derivatives**:
   - To find \( dz \), we need the partial derivatives \( \frac{\partial z}{\partial x} \) and \( \frac{\partial z}{\partial y} \).

   - **For \( \frac{\partial z}{\partial x} \)**:
     \[
     \frac{\partial z}{\partial x} = \frac{\partial}{\partial x} \left( y \sqrt{x} \right) = y \cdot \frac{1}{2\sqrt{x}}
     \]
   
   - **For \( \frac{\partial z}{\partial y} \)**:
     \[
     \frac{\partial z}{\partial y} = \frac{\partial}{\partial y} \left( y \sqrt{x} \right) = \sqrt{x}
     \]

3. **Evaluating at \( M(1, 1) \)**:
   - At the point \( M(1, 1) \), \( x = 1 \) and \( y = 1 \).
   
   - **Evaluate \( \frac{\partial z}{\partial x} \) at \( (1, 1) \)**:
     \[
     \frac{\partial z}{\partial x} \Big|_{(1, 1)} = 1 \cdot \frac{1}{2\sqrt{1}} = \frac{1}{2}
     \]
   
   - **Evaluate \( \frac{\partial z}{\partial y} \) at \( (1, 1) \)**:
     \[
     \frac{\partial z}{\partial y} \Big|_{(1, 1)} = \sqrt{1} = 1
     \]

4. **Differential \( dz \)**:
   The total differential is:
   \[
   dz = \frac{\partial z}{\partial x} dx + \frac{\partial z}{\partial y} dy
   \]
   Substituting the values at \( M(1, 1) \):
   \[
   dz = \frac{1}{2} dx + 1 \cdot dy = \frac{1}{2} dx + dy
   \]

### Correct Answer:
The correct option is:
\[
dz = 0.5dx + dy
\]

Would you like more details or have any questions?

Here are 5 relative questions to expand understanding:
1. How do you compute partial derivatives for more complex functions?
2. How does the chain rule apply to implicit functions?
3. Can you explain the geometric interpretation of differentials?
4. How would the answer change if the function involved three variables?
5. What is the significance of evaluating differentials at specific points?

**Tip**: Always check if the function can be simplified before differentiating—it can save you time and reduce errors.

Find the differential dz of the function z = √(x * y^2), evaluated at the point M(1, 1).

This page explains how to compute the differential dz of the function z = √(x * y^2), using partial derivatives and evaluating them at the point M(1, 1). It covers key calculus concepts and provides a step-by-step solution for university-level students.

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