Let 
\[h\] be a function defined for all real numbers except for 
\[-1\].
Also let 
\[h'\], the derivative of 
\[h\], be defined as 
 
\[h'(x)=\dfrac{(x+5)}{(x+1)}\].
On which intervals is 
\[h\] increasing?

To determine the intervals where the function \( h(x) \) is increasing, we analyze the derivative \( h'(x) \). A function is increasing where its derivative is positive.

Given:
\[
h'(x) = \frac{x+5}{x+1}
\]

We need to find where \( h'(x) > 0 \).

### Step 1: Analyze the Sign of \( h'(x) \)
The sign of \( h'(x) \) depends on the signs of the numerator and denominator separately.

- **Numerator:** \( x + 5 \) is positive when \( x > -5 \).
- **Denominator:** \( x + 1 \) is positive when \( x > -1 \).

### Step 2: Identify Critical Points
- At \( x = -5 \), the numerator \( x + 5 \) changes sign.
- At \( x = -1 \), the denominator \( x + 1 \) changes sign, but \( h(x) \) is not defined at \( x = -1 \).

### Step 3: Test Intervals
To determine the sign of \( h'(x) \), we test the intervals determined by the critical points:

1. **Interval \( (-\infty, -5) \):**
   - Numerator \( x + 5 < 0 \)
   - Denominator \( x + 1 < 0 \)
   - Therefore, \( h'(x) > 0 \) (positive).

2. **Interval \( (-5, -1) \):**
   - Numerator \( x + 5 > 0 \)
   - Denominator \( x + 1 < 0 \)
   - Therefore, \( h'(x) < 0 \) (negative).

3. **Interval \( (-1, \infty) \):**
   - Numerator \( x + 5 > 0 \)
   - Denominator \( x + 1 > 0 \)
   - Therefore, \( h'(x) > 0 \) (positive).

### Conclusion
The function \( h(x) \) is increasing where \( h'(x) > 0 \), which occurs in the intervals:
\[
(-\infty, -5) \cup (-1, \infty)
\]

Would you like more details, or do you have any questions?

Here are some related questions you might find helpful:
1. What is the relationship between a function's derivative and its critical points?
2. How do you determine when a function is decreasing?
3. Can a function be increasing on an interval where its derivative is zero?
4. How does the discontinuity at \( x = -1 \) affect the behavior of the function \( h(x) \)?
5. What happens to the behavior of the function \( h(x) \) near the points \( x = -5 \) and \( x = -1 \)?

**Tip:** When analyzing derivatives, always pay close attention to the points where the derivative is zero or undefined, as these often correspond to key changes in the behavior of the function.

Explore how to analyze the function h(x) and its derivative to determine where h(x) is increasing. This problem covers the derivative formula and interval analysis techniques. Suitable for advanced high school students.

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