https://r2.erweima.ai/mathbot/image/1722664490146lx56vh9h.jpg

To solve the given problem step-by-step, I'll start by extracting and interpreting the mathematical content from the image.

### Extracted Mathematical Content

From the image, we have the following equations and problem statement:

1. The given equations:
   \[
   F = x - \sqrt{x + y - \sqrt{y + z - \sqrt{z + x}}}
   \]
   \[
   F' = 1 - \frac{1}{2\sqrt{x + y - \sqrt{y + z - \sqrt{z + x}}}} \left( 1 + \frac{1}{2\sqrt{y + z - \sqrt{z + x}}} \left( 1 + \frac{1}{2\sqrt{z + x}} \right) \right)
   \]

2. Example 1:
   \[
   \text{Determine } \frac{\partial u}{\partial x_i}
   \]
   \[
   u = x_1^2 + 2x_2^2 + \ldots + nx_n^2 \implies \frac{\partial u}{\partial x_i} = 2ix_i
   \]

### Step-by-Step Solution

#### Step 1: Simplifying the Expression for \( F' \)
We need to differentiate \( F \) with respect to \( x \):

\[
F = x - \sqrt{x + y - \sqrt{y + z - \sqrt{z + x}}}
\]

Let \( g(x, y, z) = x + y - \sqrt{y + z - \sqrt{z + x}} \). Then,
\[
F = x - \sqrt{g(x, y, z)}
\]

The chain rule will be used to differentiate \( F \):
\[
\frac{dF}{dx} = 1 - \frac{1}{2\sqrt{g(x, y, z)}} \cdot \frac{dg}{dx}
\]

#### Step 2: Differentiating \( g(x, y, z) \)
\[
g(x, y, z) = x + y - \sqrt{y + z - \sqrt{z + x}}
\]

Let \( h(y, z, x) = y + z - \sqrt{z + x} \). Then,
\[
g(x, y, z) = x + y - \sqrt{h(y, z, x)}
\]

Using the chain rule,
\[
\frac{dg}{dx} = 1 - \frac{1}{2\sqrt{h(y, z, x)}} \cdot \frac{dh}{dx}
\]

#### Step 3: Differentiating \( h(y, z, x) \)
\[
h(y, z, x) = y + z - \sqrt{z + x}
\]

\[
\frac{dh}{dx} = -\frac{1}{2\sqrt{z + x}} \cdot \frac{d(z + x)}{dx} = -\frac{1}{2\sqrt{z + x}}
\]

#### Step 4: Putting it All Together
\[
\frac{dg}{dx} = 1 - \frac{1}{2\sqrt{h(y, z, x)}} \cdot \left(-\frac{1}{2\sqrt{z + x}}\right) = 1 + \frac{1}{2\sqrt{h(y, z, x)} \cdot 2\sqrt{z + x}}
\]

Substitute \( \frac{dg}{dx} \) back into the expression for \( \frac{dF}{dx} \):
\[
\frac{dF}{dx} = 1 - \frac{1}{2\sqrt{g(x, y, z)}} \left(1 + \frac{1}{2\sqrt{h(y, z, x)} \cdot 2\sqrt{z + x}}\right)
\]

### Example 1 Solution
Determine \( \frac{\partial u}{\partial x_i} \) for:
\[
u = x_1^2 + 2x_2^2 + \ldots + nx_n^2
\]

Using partial differentiation:
\[
\frac{\partial u}{\partial x_i} = 2ix_i
\]

### Summary

1. The derivative \( F' \) involves applying the chain rule multiple times due to nested square roots.
2. For the given function \( u \), the partial derivative with respect to \( x_i \) is \( 2ix_i \).

Would you like further details or have any questions?

Here are 8 related questions to expand your understanding:

1. What is the chain rule in calculus?
2. How do you differentiate nested functions?
3. What are partial derivatives and how are they used?
4. Can you explain the process of differentiating square root functions?
5. What is the significance of the derivative in mathematical analysis?
6. How do you handle implicit differentiation?
7. What are some applications of partial derivatives in real-world problems?
8. Can you show an example of differentiating a more complex function?

**Tip:** When differentiating complex functions, breaking them down into simpler parts and applying the chain rule step-by-step can make the process more manageable.

Learn how to differentiate nested square root functions step-by-step. This detailed solution covers the application of the chain rule and partial differentiation to solve complex mathematical expressions involving nested functions.

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