To find the average rate of change of the function $f(x) = \frac{1}{x+5}$ over the interval $[4, 4+h]$, we will use the formula for the average rate of change:

$\text{Average Rate of Change} = \frac{f(4+h) - f(4)}{(4+h) - 4}.$

Step 1: Find $f(4+h)$

Substitute $x = 4 + h$ into the function $f(x)$: $f(4+h) = \frac{1}{(4+h) + 5} = \frac{1}{9 + h}.$

Step 2: Find $f(4)$

Substitute $x = 4$ into the function: $f(4) = \frac{1}{4 + 5} = \frac{1}{9}.$

Step 3: Set up the formula for the average rate of change

Now, apply these values into the formula: $\text{Average Rate of Change} = \frac{\frac{1}{9+h} - \frac{1}{9}}{(4+h) - 4}.$

This simplifies to: $\text{Average Rate of Change} = \frac{\frac{1}{9+h} - \frac{1}{9}}{h}.$

Step 4: Simplify the expression

To simplify the numerator, find a common denominator between $\frac{1}{9+h}$ and $\frac{1}{9}$: $\frac{1}{9+h} - \frac{1}{9} = \frac{9 - (9+h)}{9(9+h)} = \frac{-h}{9(9+h)}.$

Thus, the expression becomes: $\text{Average Rate of Change} = \frac{\frac{-h}{9(9+h)}}{h}.$

Cancel the $h$ terms: $\text{Average Rate of Change} = \frac{-1}{9(9+h)}.$

So, the average rate of change of $f(x)$ on the interval $[4, 4+h]$ is: $\boxed{\frac{-1}{9(9+h)}}.$

Would you like a deeper explanation or further details on any step?

Here are some related questions you might find interesting:

1. What is the difference between instantaneous and average rates of change?
2. How can you find the derivative of $f(x) = \frac{1}{x+5}$?
3. How does the value of $h$ affect the average rate of change?
4. What happens to the average rate of change as $h$ approaches 0?
5. Can this method be applied to other types of functions, like polynomials or trigonometric functions?

Tip: When simplifying expressions involving fractions, always try to find a common denominator first. This helps in combining the terms easily.