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The image shows a function \( f(x) = x^2 + 3x - 8 \) and a difference quotient \( \frac{f(x+h) - f(x)}{h} \), which is used to find the derivative of the function at a point as \( h \) approaches zero.

Let's solve the difference quotient step-by-step:

1. **Substitute \( f(x) \):**
   Given \( f(x) = x^2 + 3x - 8 \):
   \[
   f(x+h) = (x+h)^2 + 3(x+h) - 8
   \]

2. **Expand \( f(x+h) \):**
   \[
   f(x+h) = (x^2 + 2xh + h^2) + (3x + 3h) - 8
   \]
   \[
   = x^2 + 2xh + h^2 + 3x + 3h - 8
   \]

3. **Difference \( f(x+h) - f(x) \):**
   \[
   f(x+h) - f(x) = (x^2 + 2xh + h^2 + 3x + 3h - 8) - (x^2 + 3x - 8)
   \]
   Simplifying, we get:
   \[
   f(x+h) - f(x) = 2xh + h^2 + 3h
   \]

4. **Form the difference quotient:**
   \[
   \frac{f(x+h) - f(x)}{h} = \frac{2xh + h^2 + 3h}{h}
   \]
   Simplifying by dividing each term by \( h \):
   \[
   = \frac{2xh}{h} + \frac{h^2}{h} + \frac{3h}{h}
   \]
   \[
   = 2x + h + 3
   \]

So, the simplified form of the difference quotient \( \frac{f(x+h) - f(x)}{h} \) is \( 2x + h + 3 \).

If we take the limit as \( h \) approaches 0, we get the derivative \( f'(x) \):
\[
f'(x) = \lim_{h \to 0} (2x + h + 3) = 2x + 3
\]

### Summary:
The difference quotient for the function \( f(x) = x^2 + 3x - 8 \) simplifies to \( 2x + h + 3 \). Taking the limit as \( h \) approaches 0, the derivative \( f'(x) \) is \( 2x + 3 \).

Would you like further details or have any questions?

### Related Questions:
1. What is the geometric interpretation of the derivative?
2. How do we apply the power rule to differentiate \( x^2 + 3x - 8 \)?
3. What is the physical meaning of the derivative in real-world problems?
4. How does the derivative relate to the slope of the tangent line at a point?
5. How can we use the derivative to find the critical points of a function?

### Tip:
Always remember that the difference quotient is the fundamental concept behind the derivative, providing the slope of the tangent line to the curve at a point.

If f(x) = x^2 + 3x - 8, then (f(x+h) - f(x))/h = ?

Learn how to compute the derivative of the function f(x) = x^2 + 3x - 8 using the difference quotient method. This example demonstrates key calculus concepts, such as limits and derivatives, and is suitable for students in advanced high school or early college calculus courses.

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