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To solve the problem, we need to calculate \( x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} + z \frac{\partial u}{\partial z} \) given that:

\[
u = \frac{y}{x} + \frac{z}{x}.
\]

### Step 1: Rewrite \( u \) in a simpler form
We can express \( u \) as:
\[
u = \frac{y + z}{x}.
\]

### Step 2: Find the partial derivatives of \( u \) with respect to \( x \), \( y \), and \( z \).

1. **Partial derivative with respect to \( x \):**
   \[
   \frac{\partial u}{\partial x} = \frac{\partial}{\partial x} \left( \frac{y + z}{x} \right) = -\frac{y + z}{x^2}.
   \]

2. **Partial derivative with respect to \( y \):**
   \[
   \frac{\partial u}{\partial y} = \frac{\partial}{\partial y} \left( \frac{y + z}{x} \right) = \frac{1}{x}.
   \]

3. **Partial derivative with respect to \( z \):**
   \[
   \frac{\partial u}{\partial z} = \frac{\partial}{\partial z} \left( \frac{y + z}{x} \right) = \frac{1}{x}.
   \]

### Step 3: Substitute into \( x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} + z \frac{\partial u}{\partial z} \)

Now, we calculate each term individually:

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

2. \( y \frac{\partial u}{\partial y} = y \cdot \frac{1}{x} = \frac{y}{x}. \)

3. \( z \frac{\partial u}{\partial z} = z \cdot \frac{1}{x} = \frac{z}{x}. \)

Adding these together:

\[
x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} + z \frac{\partial u}{\partial z} = -\frac{y + z}{x} + \frac{y}{x} + \frac{z}{x}.
\]

Simplify the expression:

\[
= -\frac{y + z}{x} + \frac{y + z}{x} = 0.
\]

### Final Answer:
\[
x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} + z \frac{\partial u}{\partial z} = 0.
\]

Would you like more details, or do you have any questions?

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Here are some related questions:

1. How do you find partial derivatives for more complex functions?
2. What is the significance of calculating \( x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} + z \frac{\partial u}{\partial z} \)?
3. Can we apply this method to functions involving exponential or trigonometric terms?
4. How does the result change if \( u \) is defined as a different ratio?
5. What are common applications of such derivative calculations in physics or engineering?

**Tip:** Simplifying expressions early can make calculations with partial derivatives more manageable.

If u = y/x + z/x, then find x * (∂u/∂x) + y * (∂u/∂y) + z * (∂u/∂z).

Learn how to calculate the expression x*(∂u/∂x) + y*(∂u/∂y) + z*(∂u/∂z) for the function u = y/x + z/x. This solution involves applying partial derivatives and simplification techniques. Suitable for undergraduate calculus students.

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