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The problem you have shared involves finding the value of \( x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} + z \frac{\partial u}{\partial z} \) where \( u = \sin^{-1} \left( \frac{x + 2y + 3z}{x^2 + y^2 + z^2} \right) \).

This expression resembles Euler's theorem for homogeneous functions, so it suggests the function's degree of homogeneity.

### Step-by-step approach:

1. **Expression for \( u \):**
   \[
   u = \sin^{-1} \left( \frac{x + 2y + 3z}{x^2 + y^2 + z^2} \right)
   \]
   Let’s denote the argument inside the inverse sine as:
   \[
   f(x, y, z) = \frac{x + 2y + 3z}{x^2 + y^2 + z^2}
   \]

2. **Find Partial Derivatives:**
   We need to compute \( \frac{\partial u}{\partial x}, \frac{\partial u}{\partial y}, \frac{\partial u}{\partial z} \), which can be found using the chain rule and implicit differentiation since \( u = \sin^{-1}(f) \).

3. **Apply the formula:**
   We are asked to evaluate:
   \[
   x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} + z \frac{\partial u}{\partial z}
   \]
   Using Euler's theorem on homogeneous functions, it suggests a simplification related to the degree of the function inside the sine inverse.

I'll compute this step-by-step.The value of the expression \( x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} + z \frac{\partial u}{\partial z} \) simplifies to:

\[
-\frac{x + 2y + 3z}{\sqrt{\frac{x^4 + 2x^2y^2 + 2x^2z^2 - x^2 - 4xy - 6xz + y^4 + 2y^2z^2 - 4y^2 - 12yz + z^4 - 9z^2}{x^4 + 2x^2y^2 + 2x^2z^2 + y^4 + 2y^2z^2 + z^4}} \cdot (x^2 + y^2 + z^2)}
\]

This is quite a complex expression, but the key insight is that it evaluates the directional derivative of the function \( u \) along the vector \( (x, y, z) \). Further simplification could involve specific values of \( x, y, z \), depending on your context.

Would you like more details on how the partial derivatives were computed, or on any other step?

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Here are 5 related questions that could help expand your understanding:
1. What is the geometric interpretation of this expression in terms of gradients and directional derivatives?
2. How would the result change if the function \( u \) were homogeneous of a different degree?
3. Can this expression be simplified for specific values of \( x, y, z \)?
4. How would the result differ if a different function of \( x, y, z \) were used instead of \( u \)?
5. How do we apply Euler's theorem to homogeneous functions in this context?

**Tip:** When differentiating complex multivariable functions, applying the chain rule carefully can lead to important simplifications based on the problem's symmetry.

If u = sin^(-1)((x + 2y + 3z) / (x^2 + y^2 + z^2)), find the value of x(∂u/∂x) + y(∂u/∂y) + z(∂u/∂z).

Learn how to compute the value of x(∂u/∂x) + y(∂u/∂y) + z(∂u/∂z) where u = sin^(-1)((x + 2y + 3z) / (x^2 + y^2 + z^2)). This problem involves using partial derivatives and Euler's theorem on homogeneous functions. Suitable for advanced calculus students.

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