$$ \sqrt{x+\sqrt{x+\sqrt{x}}} $$findderivative

To find the derivative of the expression 

\[
f(x) = \sqrt{x + \sqrt{x + \sqrt{x}}}
\]

we'll go through it step-by-step.

### Step 1: Rewrite the Expression in Terms of Nested Functions
Define each inner function to simplify differentiation:

1. Let $ u = \sqrt{x + \sqrt{x}} $.
2. Then, the entire function becomes $ f(x) = \sqrt{x + u} $.

### Step 2: Differentiate $ f(x) $
Using the chain rule, we differentiate $ f(x) = \sqrt{x + u} $ with respect to $ x $:

\[
f'(x) = \frac{1}{2\sqrt{x + u}} \cdot \left(1 + \frac{du}{dx}\right)
\]

### Step 3: Differentiate $ u $ with Respect to $ x $
Recall that $ u = \sqrt{x + \sqrt{x}} $.

Using the chain rule again:

\[
\frac{du}{dx} = \frac{1}{2\sqrt{x + \sqrt{x}}} \cdot \left(1 + \frac{d}{dx}\sqrt{x}\right)
\]

### Step 4: Differentiate the Innermost Square Root
Since $\sqrt{x} = x^{1/2}$, we have:

\[
\frac{d}{dx}\sqrt{x} = \frac{1}{2\sqrt{x}}
\]

### Step 5: Substitute and Simplify
Now, putting everything together:

\[
\frac{du}{dx} = \frac{1}{2\sqrt{x + \sqrt{x}}} \cdot \left(1 + \frac{1}{2\sqrt{x}}\right)
\]

And substitute this expression for $ \frac{du}{dx} $ back into the expression for $ f'(x) $:

\[
f'(x) = \frac{1}{2\sqrt{x + \sqrt{x + \sqrt{x}}}} \cdot \left(1 + \frac{1}{2\sqrt{x + \sqrt{x}}} \cdot \left(1 + \frac{1}{2\sqrt{x}}\right)\right)
\]

This gives the derivative of $ f(x) = \sqrt{x + \sqrt{x + \sqrt{x}}} $. 

Would you like me to expand on any of the steps or simplify further?

---

### Follow-up Questions
1. How would this process change if there were more nested square roots?
2. What are the implications of this derivative at different values of $ x $, especially near 0?
3. How can we use this derivative to find local maxima or minima for this function?
4. What would the second derivative of this expression look like?
5. How would the differentiation process change if we were differentiating $ \sqrt{x + 2\sqrt{x + 3\sqrt{x}}} $?

### Tip
When differentiating nested functions, use substitution to organize and apply the chain rule step-by-step for clarity.

Discover how to find the derivative of the nested square root expression $ \sqrt{x + \sqrt{x + \sqrt{x}}} $ with a detailed solution. This problem explores advanced differentiation concepts, including the chain rule applied to nested functions, and is ideal for high school calculus students.

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