The problem you're working on asks you to find the difference quotient for the function $f(x) = x^2 + 9$. The difference quotient is a way to approximate the derivative of a function and is given by the formula:

$\frac{f(x+h) - f(x)}{h}$ where $h \neq 0$.

Step-by-step solution:

1. Substitute $f(x) = x^2 + 9$ into the formula: $f(x+h) = (x+h)^2 + 9$

2. Expand $f(x+h)$: $(x+h)^2 + 9 = x^2 + 2xh + h^2 + 9$

3. Now subtract $f(x)$: $f(x+h) - f(x) = (x^2 + 2xh + h^2 + 9) - (x^2 + 9)$

4. Simplify the expression: $f(x+h) - f(x) = x^2 + 2xh + h^2 + 9 - x^2 - 9$ $= 2xh + h^2$

5. Now divide by $h$: $\frac{f(x+h) - f(x)}{h} = \frac{2xh + h^2}{h}$ $= 2x + h$

So, the difference quotient is:

$\boxed{2x + h}$

Would you like further details or have any other questions?

Here are 5 related questions to deepen your understanding:

1. How would the difference quotient change if $f(x) = x^3 + 1$?
2. What does the difference quotient represent in terms of calculus concepts?
3. How can you use the difference quotient to find the derivative of a function?
4. What happens to the difference quotient as $h \to 0$?
5. Why is it important to ensure $h \neq 0$ in the difference quotient formula?

Tip: The difference quotient is essentially the definition of the derivative, and as $h$ approaches zero, it becomes the derivative at a point!