To find the derivative of the function $y = \sqrt{x^2 + 5}$, we will use the chain rule.

1. Rewrite the function: $y = (x^2 + 5)^{\frac{1}{2}}$

2. Now apply the chain rule. The general form is: $\frac{d}{dx} \left[ (u(x))^n \right] = n(u(x))^{n-1} \cdot \frac{du(x)}{dx}$

   Here, $u(x) = x^2 + 5$ and $n = \frac{1}{2}$.

3. Differentiate using the chain rule: $\frac{dy}{dx} = \frac{1}{2}(x^2 + 5)^{-\frac{1}{2}} \cdot \frac{d}{dx}(x^2 + 5)$

4. Now differentiate $x^2 + 5$: $\frac{d}{dx}(x^2 + 5) = 2x$

5. Substituting it into the chain rule expression: $\frac{dy}{dx} = \frac{1}{2}(x^2 + 5)^{-\frac{1}{2}} \cdot 2x$

6. Simplify: $\frac{dy}{dx} = \frac{x}{\sqrt{x^2 + 5}}$

Thus, the derivative of $y = \sqrt{x^2 + 5}$ is: $\frac{dy}{dx} = \frac{x}{\sqrt{x^2 + 5}}$

Would you like further details on this solution or have any questions?

Here are some related questions to expand your understanding:

1. How do you apply the chain rule to more complex functions?
2. What is the derivative of $y = \sqrt{x^2 - 4x + 7}$?
3. Can you explain how to differentiate powers of binomials?
4. How does the chain rule apply to trigonometric functions?
5. How do we find the second derivative of $y = \sqrt{x^2 + 5}$?

Tip: When applying the chain rule, first identify the "outer" and "inner" functions before differentiating.