The given function is \( g(x) = x \tan^{-1}(x + 1) \), where \( \tan^{-1}(x + 1) \) refers to the inverse tangent (or arctan) of \( x + 1 \). Let's analyze and find its properties.

### Differentiating \( g(x) \) with respect to \( x \):

To find the derivative \( g'(x) \), we'll use the product rule, which states:

\[
\frac{d}{dx}[u \cdot v] = u' \cdot v + u \cdot v'
\]

Here, let:
- \( u = x \) and \( v = \tan^{-1}(x + 1) \).

#### Step 1: Differentiate \( u = x \)

\[
u' = 1
\]

#### Step 2: Differentiate \( v = \tan^{-1}(x + 1) \)

Using the derivative of \( \tan^{-1}(u) \), which is \( \frac{d}{du} \tan^{-1}(u) = \frac{1}{1 + u^2} \), we get:

\[
\frac{d}{dx} \tan^{-1}(x + 1) = \frac{1}{1 + (x + 1)^2}
\]

#### Step 3: Apply the product rule

\[
g'(x) = 1 \cdot \tan^{-1}(x + 1) + x \cdot \frac{1}{1 + (x + 1)^2}
\]

Thus,

\[
g'(x) = \tan^{-1}(x + 1) + \frac{x}{1 + (x + 1)^2}
\]

This is the derivative of \( g(x) \).

### Want more details or have any questions? 
Below are 5 questions related to this information:

1. How would you find the second derivative of \( g(x) \)?
2. What is the domain and range of \( g(x) \)?
3. How can you find the critical points of \( g(x) \)?
4. How would you determine if \( g(x) \) is increasing or decreasing?
5. Can you find the limit of \( g(x) \) as \( x \to \infty \)?

**Tip:** When differentiating composite functions like \( \tan^{-1}(x + 1) \), be mindful of the chain rule which captures how changes in the inner function propagate through the derivative.

Learn how to find the derivative of g(x) = x * tan^-1(x + 1) using the product rule and chain rule. This solution explains the step-by-step process, applying key calculus concepts such as inverse trigonometric functions and differentiation techniques. Suitable for advanced high school or early college students.

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