To solve this problem, let’s find the derivative $\frac{dy}{dx}$ for the function given in the problem:

Problem

Given: $y = \sqrt{x + \sqrt{1 + x}}$

We need to find $\frac{dy}{dx}$.

Solution

To differentiate $y = \sqrt{x + \sqrt{1 + x}}$, we will use the chain rule step-by-step.

1. Rewrite $y$: Let: $y = (x + \sqrt{1 + x})^{1/2}$

2. Differentiate $y$ with respect to $x$: Using the chain rule, we have: $\frac{dy}{dx} = \frac{1}{2} (x + \sqrt{1 + x})^{-1/2} \cdot \frac{d}{dx}(x + \sqrt{1 + x})$

3. Differentiate the inner function $x + \sqrt{1 + x}$: $\frac{d}{dx}(x + \sqrt{1 + x}) = 1 + \frac{1}{2} (1 + x)^{-1/2}$

4. Combine the results: Substitute back into our expression for $\frac{dy}{dx}$: $\frac{dy}{dx} = \frac{1}{2} (x + \sqrt{1 + x})^{-1/2} \cdot \left(1 + \frac{1}{2} (1 + x)^{-1/2}\right)$

Final Answer

$\frac{dy}{dx} = \frac{1 + \frac{1}{2} (1 + x)^{-1/2}}{2 \sqrt{x + \sqrt{1 + x}}}$

This is the derivative $\frac{dy}{dx}$ for the given function.

Would you like a detailed step-by-step explanation of any part?

Here are five related questions for further practice:

1. How would you approach differentiating a function with nested square roots in general?
2. What alternative methods could you use to verify the derivative result here?
3. Can you evaluate $\frac{dy}{dx}$ at specific values of $x$, like $x = 0$ or $x = 1$?
4. How would you differentiate if there were an additional term outside the square root?
5. What are the implications of such derivatives in physics or engineering contexts?

Tip: Always break down complex derivatives into manageable parts, especially when using the chain rule on nested functions.