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The image contains the following problem:

**If \( e^u = x^2 + y^2 + z^2 \), prove that:**

\[
\frac{x u_x}{y z} = \frac{y u_y}{z x} = \frac{z u_z}{x y}
\]

Where \( u_x \), \( u_y \), and \( u_z \) refer to the partial derivatives of \( u \) with respect to \( x \), \( y \), and \( z \), respectively.

### Solution outline:

1. **Given equation:**  
   \( e^u = x^2 + y^2 + z^2 \)

2. **Take partial derivatives with respect to \( x \), \( y \), and \( z \):**
   - Differentiating both sides with respect to \( x \), \( y \), and \( z \), respectively, using the chain rule.
   - For example, differentiating \( e^u \) with respect to \( x \) gives:
     \[
     e^u \cdot u_x = 2x
     \]
     Thus, the partial derivative \( u_x \) can be solved as:
     \[
     u_x = \frac{2x}{e^u}
     \]
     Similarly, we differentiate with respect to \( y \) and \( z \).

3. **Substitute and simplify the given expression:**  
   Using the partial derivatives obtained, substitute them back into the expression:
   \[
   \frac{x u_x}{yz}, \frac{y u_y}{zx}, \frac{z u_z}{xy}
   \]
   Then, verify that these ratios are indeed equal.

Would you like me to proceed with the detailed derivation of these steps?

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### 5 Related Questions:
1. What is the role of partial derivatives in multivariable calculus?
2. How can you differentiate exponential functions with respect to multiple variables?
3. Why is the chain rule crucial when differentiating functions of several variables?
4. How do we solve systems of equations involving partial derivatives?
5. What are the practical applications of this type of mathematical problem in physics or engineering?

### Tip:
When dealing with exponential functions and their derivatives, remember that the chain rule helps link the inside function and the exponent.

If e^u = x^2 + y^2 + z^2, prove that xu_x / yz = yu_y / zx = zu_z / xy.

Learn how to prove the relationship between partial derivatives for the equation e^u = x^2 + y^2 + z^2. This solution involves using the chain rule and simplifying ratios of partial derivatives. Suitable for students studying multivariable calculus.

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