find the derivative of the following function: y=(x^2+5)^1/2

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.

Find the derivative of the function y = sqrt(x^2 + 5) using the chain rule. This solution includes a step-by-step explanation of applying the chain rule, identifying inner and outer functions, and simplifying the result. Ideal for calculus students in Grades 11-12 or early college.

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