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To solve **question 2, problem 5**, let's first clarify the problem:

The expression is:
\[
f(x) = \sqrt{x^2 + 4x} - \sqrt{(x - 1)^2}
\]

We are tasked with finding the first derivative of this function, \( f'(x) \).

### Step-by-Step Solution:

#### 1. Simplify the Expression:

The second square root simplifies as follows:
\[
\sqrt{(x - 1)^2} = |x - 1|
\]
Thus, the function becomes:
\[
f(x) = \sqrt{x^2 + 4x} - |x - 1|
\]

#### 2. Find the Derivative of Each Part:

1. **Derivative of** \( \sqrt{x^2 + 4x} \):
   Using the chain rule, let \( u(x) = x^2 + 4x \), so:
   \[
   \frac{d}{dx} \sqrt{u(x)} = \frac{1}{2\sqrt{u(x)}} \cdot u'(x)
   \]
   The derivative of \( u(x) = x^2 + 4x \) is \( u'(x) = 2x + 4 \). Thus, the derivative is:
   \[
   \frac{d}{dx} \sqrt{x^2 + 4x} = \frac{2x + 4}{2\sqrt{x^2 + 4x}} = \frac{x + 2}{\sqrt{x^2 + 4x}}
   \]

2. **Derivative of** \( |x - 1| \):
   The derivative of \( |x - 1| \) depends on the value of \( x \):
   - For \( x > 1 \), \( |x - 1| = x - 1 \), so \( \frac{d}{dx} |x - 1| = 1 \).
   - For \( x < 1 \), \( |x - 1| = -(x - 1) = 1 - x \), so \( \frac{d}{dx} |x - 1| = -1 \).

#### 3. Write the Final Derivative:

For \( x > 1 \), the derivative is:
\[
f'(x) = \frac{x + 2}{\sqrt{x^2 + 4x}} - 1
\]
For \( x < 1 \), the derivative is:
\[
f'(x) = \frac{x + 2}{\sqrt{x^2 + 4x}} + 1
\]

### Conclusion:
The derivative of the function depends on whether \( x \) is greater than or less than 1, as the absolute value function introduces a piecewise condition.

Would you like further details on any specific part? Here are some related questions:

1. How do you apply the chain rule to other functions like this?
2. What is the significance of absolute values in derivatives?
3. Can we simplify the expression for easier differentiation?
4. How do we graph this piecewise derivative function?
5. How does the behavior change around \( x = 1 \) due to the absolute value?

**Tip**: When dealing with square roots or absolute values, always check for domains or conditions that can affect the function’s behavior!

Learn how to solve for the derivative of f(x) = √(x² + 4x) - |x - 1| using the chain rule and piecewise differentiation. This solution involves the derivatives of square roots and absolute value functions, key concepts in calculus for college-level statistics and economics courses.

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