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The problem in the image asks to calculate the derivative of the function \( f(x) = 9x^2 + 7x + 8 \) at \( x = -8 \) using the definition of the derivative.

### Step-by-Step Solution:

The derivative of a function at a point \( a \), using the definition of the derivative, is given by:
\[
f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}
\]
For this problem, \( a = -8 \) and \( f(x) = 9x^2 + 7x + 8 \).

1. **Evaluate \( f(-8) \):**
\[
f(-8) = 9(-8)^2 + 7(-8) + 8 = 9(64) - 56 + 8 = 576 - 56 + 8 = 528
\]

2. **Substitute into the definition:**
\[
f'(-8) = \lim_{x \to -8} \frac{9x^2 + 7x + 8 - 528}{x + 8}
\]
Simplifying the numerator:
\[
9x^2 + 7x + 8 - 528 = 9x^2 + 7x - 520
\]
Thus, the expression becomes:
\[
f'(-8) = \lim_{x \to -8} \frac{9x^2 + 7x - 520}{x + 8}
\]

3. **Factor the numerator:**
We need to factor \( 9x^2 + 7x - 520 \). After factoring, we get:
\[
9x^2 + 7x - 520 = (x + 8)(9x - 65)
\]
Thus, the limit expression becomes:
\[
f'(-8) = \lim_{x \to -8} \frac{(x + 8)(9x - 65)}{x + 8}
\]

4. **Cancel the common factor \( (x + 8) \):**
\[
f'(-8) = \lim_{x \to -8} (9x - 65)
\]

5. **Evaluate the limit:**
\[
f'(-8) = 9(-8) - 65 = -72 - 65 = -137
\]

### Conclusion:
The value of the limit, and thus the derivative \( f'(-8) \), is indeed **-137**.

Would you like further details or have any questions?

Here are five related questions:
1. What is the general formula for the derivative of a quadratic function?
2. How would you find the derivative of \( f(x) = 9x^2 + 7x + 8 \) using standard derivative rules?
3. What is the geometric interpretation of the derivative at a point?
4. How does the derivative change as the degree of the polynomial increases?
5. How can we check our answer by using a derivative calculator?

**Tip**: Always check if the function can be factored to simplify the calculation of limits.

Let f(x) = 9x^2 + 7x + 8. Then according to the definition of derivative f'(-8) is the limit as x tends to -8 of the expression. The value of this limit is -137.

Learn how to calculate the derivative of the quadratic function f(x) = 9x^2 + 7x + 8 at x = -8 using the limit definition. This detailed step-by-step solution covers key concepts in calculus such as limits and the definition of the derivative. Suitable for high school students and early university learners.

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