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To find the derivative of the function \( f(x) = 9x + 11 \) using the definition of the derivative, we use the following formula:

\[
f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}
\]

### Step-by-Step Solution:

1. **Find \( f(x+h) \):**
   \[
   f(x+h) = 9(x+h) + 11 = 9x + 9h + 11
   \]

2. **Substitute into the definition:**
   \[
   f'(x) = \lim_{h \to 0} \frac{(9x + 9h + 11) - (9x + 11)}{h}
   \]
   Simplify the numerator:
   \[
   f'(x) = \lim_{h \to 0} \frac{9x + 9h + 11 - 9x - 11}{h} = \lim_{h \to 0} \frac{9h}{h}
   \]

3. **Simplify the expression:**
   \[
   f'(x) = \lim_{h \to 0} 9 = 9
   \]

Thus, the derivative of \( f(x) = 9x + 11 \) is:

\[
f'(x) = 9
\]

Would you like further explanation or details on the concept of derivatives?

Here are 5 related questions to explore:
1. How do you compute the derivative of a quadratic function using the definition?
2. What is the geometric interpretation of the derivative?
3. How is the derivative of a constant function computed?
4. Can the definition of the derivative be used to find second derivatives?
5. How does the derivative relate to the slope of a tangent line at a given point?

**Tip:** When differentiating linear functions, the derivative is simply the coefficient of \( x \) since the rate of change is constant.

Use the definition of the derivative to find the derivative of f(x) = 9x + 11.

Learn how to compute the derivative of the linear function f(x) = 9x + 11 using the definition of the derivative. This detailed solution explains the steps using limits and basic calculus concepts, suitable for students in Grades 11-12.

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