The difference quotient for a function $f(x)$ is given by:

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

Step 1: Given function

The given function is $f(x) = 7x$.

Step 2: Find $f(x+h)$

Substitute $x+h$ into the function $f(x)$:

$f(x+h) = 7(x+h) = 7x + 7h$

Step 3: Compute the difference quotient

Now substitute $f(x+h)$ and $f(x)$ into the difference quotient formula:

$\frac{f(x+h) - f(x)}{h} = \frac{(7x + 7h) - 7x}{h}$

Step 4: Simplify the expression

$\frac{7x + 7h - 7x}{h} = \frac{7h}{h}$

Since $h \neq 0$, the $h$'s cancel out:

$\frac{7h}{h} = 7$

Final Answer:

The simplified difference quotient is:

$7$

Would you like more details, or do you have any questions?

Here are some related questions to explore:

1. What is the difference quotient for $f(x) = x^2$?
2. How does the difference quotient relate to the derivative?
3. What happens to the difference quotient as $h \to 0$?
4. Can you find the difference quotient for $f(x) = 3x^3$?
5. How do you apply the difference quotient to functions with multiple terms?

Tip: The difference quotient is the foundation of finding the derivative of a function!