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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?

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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!

Find the difference quotient of f(x) = x^2 + 9.

Learn how to calculate the difference quotient of the function f(x) = x^2 + 9. This step-by-step solution demonstrates key algebraic principles like function expansion and simplifying expressions. Suitable for high school students in pre-calculus and calculus courses.

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