Use f prime left parenthesis x right parenthesis equals ModifyingBelow lim With h right arrow 0 StartFraction f left parenthesis x plus h right parenthesis minus f left parenthesis x right parenthesis Over h EndFraction
to find the derivative at x for the given function.
s left parenthesis x right parenthesis equals 8 x plus 7

To find the derivative of the function \( s(x) = 8x + 7 \) using the definition of a derivative, we can apply the following formula:

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

### Step-by-Step Solution:

1. **Start with the given function:**
   \[
   s(x) = 8x + 7
   \]

2. **Substitute \( s(x+h) \):**
   \[
   s(x+h) = 8(x+h) + 7 = 8x + 8h + 7
   \]

3. **Now, apply the definition of the derivative:**
   \[
   s'(x) = \lim_{h \to 0} \frac{(8x + 8h + 7) - (8x + 7)}{h}
   \]

4. **Simplify the numerator:**
   \[
   s'(x) = \lim_{h \to 0} \frac{8x + 8h + 7 - 8x - 7}{h} = \lim_{h \to 0} \frac{8h}{h}
   \]

5. **Simplify further:**
   \[
   s'(x) = \lim_{h \to 0} 8
   \]

6. **Since the limit of a constant is the constant itself:**
   \[
   s'(x) = 8
   \]

### Conclusion:

The derivative of the function \( s(x) = 8x + 7 \) is \( s'(x) = 8 \).

Do you want further details or have any questions?

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Here are 5 related questions for further practice:
1. What is the derivative of \( f(x) = 3x^2 + 5x - 2 \) using the definition of a derivative?
2. How do you find the derivative of \( g(x) = 10x^3 \) using limits?
3. What is the physical interpretation of a derivative?
4. How do you find the second derivative of a function?
5. What is the derivative of \( h(x) = e^x \) using the limit definition?

**Tip:** When calculating derivatives using the limit definition, simplify the expressions step-by-step to avoid errors.

Use f prime left parenthesis x right parenthesis equals ModifyingBelow lim With h right arrow 0 StartFraction f left parenthesis x plus h right parenthesis minus f left parenthesis x right parenthesis Over h EndFraction to find the derivative at x for the given function. s left parenthesis x right parenthesis equals 8 x plus 7

Explore how to find the derivative of the linear function s(x) = 8x + 7 using the limit definition. This step-by-step guide walks through applying the derivative formula and simplifying the expressions. Suitable for students in Grades 11-12, this problem covers core calculus concepts like limits and derivatives.

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