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The problem involves using the four-step process to find the slope of the tangent line (i.e., the derivative) for the function \( f(x) = 9 - 7x \).

### The four steps of the limit definition of the derivative are as follows:

1. **Step 1:** Find \( f(x + h) \).
   
   \( f(x + h) = 9 - 7(x + h) = 9 - 7x - 7h \)

2. **Step 2:** Compute \( f(x + h) - f(x) \).
   
   \( f(x + h) - f(x) = (9 - 7x - 7h) - (9 - 7x) = -7h \)

3. **Step 3:** Divide by \( h \).
   
   \( \frac{f(x + h) - f(x)}{h} = \frac{-7h}{h} = -7 \)

4. **Step 4:** Take the limit as \( h \to 0 \).
   
   \( f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} = \lim_{h \to 0} (-7) = -7 \)

### Conclusion:
The slope of the tangent line is \( f'(x) = -7 \).

Let me know if you'd like further clarification or have any questions!

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Here are 5 related questions:

1. What is the geometric interpretation of the derivative of a linear function like \( f(x) = 9 - 7x \)?
2. How does the derivative change if the function involves higher powers of \( x \)?
3. Can you apply the same four-step process to find the derivative of a quadratic function like \( f(x) = x^2 \)?
4. How would the result change if you used different values for the constants in \( f(x) = 9 - 7x \)?
5. What happens to the slope of the tangent if you modify the function to include additional terms, like \( f(x) = 9 - 7x + x^2 \)?

**Tip**: When computing derivatives, always check if the function is linear, as its derivative is constant across all points!

Use the four-step process to find the slope of the tangent line to the graph of the function f(x) = 9 - 7x.

Learn how to find the slope of the tangent line to the graph of the function f(x) = 9 - 7x using the four-step process of the derivative definition. This step-by-step guide covers calculus concepts such as limits and derivatives, suitable for high school students in Grades 11-12.

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