To solve the problem:

We are tasked to find the difference quotient for the given function:

$\frac{f(x+h) - f(x)}{h}, \quad h \neq 0,$

where $f(x) = 3x + 6$.

Step-by-Step Solution:

1. Substitute $f(x+h)$: $f(x+h) = 3(x+h) + 6 = 3x + 3h + 6.$

2. Find $f(x+h) - f(x)$: $f(x+h) - f(x) = \big(3x + 3h + 6\big) - \big(3x + 6\big).$ Simplify: $f(x+h) - f(x) = 3h.$

3. Divide by $h$: $\frac{f(x+h) - f(x)}{h} = \frac{3h}{h}.$ Simplify (since $h \neq 0$): $\frac{f(x+h) - f(x)}{h} = 3.$

Final Answer:

The difference quotient for $f(x) = 3x + 6$ is: $\boxed{3}.$

Let me know if you need further clarification or additional steps explained!

Related Questions:

1. What is the physical meaning of the difference quotient in calculus?
2. How does the result $3$ relate to the derivative of $f(x)$?
3. What happens to the difference quotient as $h \to 0$?
4. How would the process change if $f(x) = x^2$?
5. What are some applications of the difference quotient in real-world scenarios?

Tip:

The difference quotient is a foundational concept in calculus, as it leads directly to the definition of a derivative!