To find the derivative of the function \( f(x) = \sqrt{x^2 + x} \), we'll use the chain rule and basic differentiation rules.

### Step 1: Rewrite the function
Rewrite the function as a power for easier differentiation:
\[
f(x) = (x^2 + x)^{1/2}
\]

### Step 2: Apply the chain rule
The chain rule states that if \( f(x) = g(h(x)) \), then \( f'(x) = g'(h(x)) \cdot h'(x) \).

In this case, let \( g(u) = u^{1/2} \) and \( u = x^2 + x \). So, we'll first differentiate the outer function and then multiply by the derivative of the inner function.

1. The derivative of \( g(u) = u^{1/2} \) is:
   \[
   g'(u) = \frac{1}{2}u^{-1/2}
   \]
   Substituting \( u = x^2 + x \), this becomes:
   \[
   \frac{1}{2}(x^2 + x)^{-1/2}
   \]

2. Now, differentiate the inner function \( h(x) = x^2 + x \):
   \[
   h'(x) = 2x + 1
   \]

### Step 3: Combine the results
Using the chain rule, we multiply the derivative of the outer function by the derivative of the inner function:
\[
f'(x) = \frac{1}{2}(x^2 + x)^{-1/2} \cdot (2x + 1)
\]

### Step 4: Simplify
Thus, the derivative of \( f(x) = \sqrt{x^2 + x} \) is:
\[
f'(x) = \frac{2x + 1}{2\sqrt{x^2 + x}}
\]

Let me know if you want further details or have any questions.

### Related questions:
1. How do you apply the chain rule in more complex functions?
2. Can you find the second derivative of this function?
3. What is the geometric interpretation of the derivative of a square root function?
4. How do you differentiate products of functions with square roots?
5. How can you apply the quotient rule to functions involving roots?

### Tip:
Always simplify the function before differentiating when possible, as it can help you avoid unnecessary complications.

Discover how to find the derivative of the function sqrt(x^2 + x) using the chain rule. This step-by-step guide simplifies the process of differentiation, perfect for students learning calculus at a high school or college level.

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