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We are tasked with finding the derivative of the given function:

\[
y(x) = \sqrt{x^2 + 1}
\]

### Step-by-step process:

1. **Rewrite the square root in exponent form:**
   \[
   y(x) = (x^2 + 1)^{1/2}
   \]

2. **Differentiate using the chain rule:**
   The chain rule states that:
   \[
   \frac{d}{dx} [f(g(x))] = f'(g(x)) \cdot g'(x)
   \]
   Here, \( f(u) = u^{1/2} \) and \( g(x) = x^2 + 1 \). We need to differentiate both parts.

   - Derivative of the outer function \( f(u) = u^{1/2} \):
     \[
     \frac{d}{du} [u^{1/2}] = \frac{1}{2} u^{-1/2}
     \]

   - Derivative of the inner function \( g(x) = x^2 + 1 \):
     \[
     \frac{d}{dx} [x^2 + 1] = 2x
     \]

3. **Apply the chain rule:**
   \[
   \frac{dy}{dx} = \frac{1}{2} (x^2 + 1)^{-1/2} \cdot 2x
   \]

4. **Simplify the expression:**
   The 2's cancel out:
   \[
   \frac{dy}{dx} = \frac{x}{\sqrt{x^2 + 1}}
   \]

Thus, the derivative is:

\[
\frac{dy}{dx} = \frac{x}{\sqrt{x^2 + 1}}
\]

Let me know if you would like more details or have any questions!

### Related questions:
1. How would you differentiate \( y = \sqrt{2x + 3} \)?
2. Can you explain the chain rule in more detail?
3. What is the derivative of \( y = \ln(x^2 + 1) \)?
4. How do you apply the product rule in differentiation?
5. What happens when differentiating functions involving trigonometric terms?

### Tip:
Always check if rewriting expressions (like square roots) in exponent form can make differentiation easier.

Find the derivative of y(x) = √(x^2 + 1).

Learn how to find the derivative of the function y(x) = √(x^2 + 1) using the chain rule. Step-by-step solution provided, applying key calculus concepts such as the chain rule and power rule. Suitable for Grade 12 or college-level calculus students.

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