This expression represents a difference quotient, which is commonly used in calculus to find the derivative of a function. Here, the given expression can be analyzed and simplified step by step.

The provided expression is:

$\frac{\frac{8}{a+h+1} - \frac{8}{a+1}}{h}$

Let's simplify it step by step:

Step 1: Combine the fractions in the numerator

The numerator is: $\frac{8}{a+h+1} - \frac{8}{a+1}$ To combine these fractions, find the common denominator, which is $(a+h+1)(a+1)$. This gives: $\frac{8(a+1) - 8(a+h+1)}{(a+h+1)(a+1)}$

Simplify the numerator: $8(a+1) - 8(a+h+1) = 8a + 8 - 8a - 8h - 8 = -8h$

So the combined fraction becomes: $\frac{-8h}{(a+h+1)(a+1)}$

Step 2: Divide by $h$

The entire expression becomes: $\frac{\frac{-8h}{(a+h+1)(a+1)}}{h}$

This is equivalent to multiplying by the reciprocal of $h$: $\frac{-8h}{(a+h+1)(a+1)} \cdot \frac{1}{h} = \frac{-8}{(a+h+1)(a+1)}$

Final Answer:

The simplified expression is: $\frac{-8}{(a+h+1)(a+1)}$

Would you like me to explain a specific step further or analyze related concepts?

Related Questions:

1. What is the derivative of $f(x) = \frac{8}{x+1}$ using the definition of a derivative?
2. How does the difference quotient relate to the slope of a tangent line?
3. What happens to the expression as $h \to 0$?
4. How can this simplification be visualized geometrically?
5. What are the conditions under which this function is undefined?

Tip:

Always verify if the variable $h$ can approach 0 without making the denominator undefined when simplifying limits in calculus.