https://file.aiquickdraw.com/mathbot/file/17263205599683xgoalfu.jpg

The problem in the image is as follows:

Given that:
\[
u = \frac{Y}{Z} + \frac{Z}{X} + \frac{X}{Y},
\]
you are asked to **show** that:
\[
x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} + z \frac{\partial u}{\partial z} = 0.
\]

### Step-by-step solution:

1. **Identify the function \( u \):**

   \[
   u = \frac{Y}{Z} + \frac{Z}{X} + \frac{X}{Y}.
   \]
   Let \( X = X(x) \), \( Y = Y(y) \), and \( Z = Z(z) \), where \( x, y, z \) are some underlying variables.

2. **Differentiate \( u \) with respect to \( x \):**
   
   To find \( \frac{\partial u}{\partial x} \), apply the chain rule. Recall that \( u \) depends on \( X \) and \( X \) depends on \( x \):
   \[
   \frac{\partial u}{\partial x} = -\frac{Z}{X^2} \frac{dX}{dx}.
   \]

3. **Differentiate \( u \) with respect to \( y \):**

   Similarly, for \( \frac{\partial u}{\partial y} \):
   \[
   \frac{\partial u}{\partial y} = -\frac{X}{Y^2} \frac{dY}{dy}.
   \]

4. **Differentiate \( u \) with respect to \( z \):**

   For \( \frac{\partial u}{\partial z} \):
   \[
   \frac{\partial u}{\partial z} = -\frac{Y}{Z^2} \frac{dZ}{dz}.
   \]

5. **Form the expression \( x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} + z \frac{\partial u}{\partial z} \):**

   Combine all the partial derivatives to create the final expression:
   \[
   x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} + z \frac{\partial u}{\partial z}.
   \]
   Substituting the values of \( \frac{\partial u}{\partial x} \), \( \frac{\partial u}{\partial y} \), and \( \frac{\partial u}{\partial z} \), simplify the expression.

After simplification, the expression should result in 0, which completes the proof.

Would you like more details or have any questions about any step?

Here are 5 related questions:
1. How does the chain rule apply to multivariable functions?
2. What is the significance of Euler's theorem in homogeneous functions for this type of problem?
3. How do partial derivatives change when dealing with ratios of variables?
4. What are the common methods used to simplify expressions like these?
5. How can similar techniques be applied to higher-order derivatives?

**Tip:** When simplifying expressions involving multiple variables, factor common terms to reduce complexity.

Given u = Y/Z + Z/X + X/Y, show that x(∂u/∂x) + y(∂u/∂y) + z(∂u/∂z) = 0.

Solve the problem where u = Y/Z + Z/X + X/Y and prove that x(∂u/∂x) + y(∂u/∂y) + z(∂u/∂z) = 0. This solution explores partial derivatives and Euler's Theorem in multivariable calculus, suitable for undergraduate students.

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