To find the difference quotient for the function $f(x) = 7x + 1$, we use the formula:

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

Step 1: Find $f(x+h)$

Start by substituting $x+h$ into the function $f(x)$:

$f(x+h) = 7(x+h) + 1$

Simplify:

$f(x+h) = 7x + 7h + 1$

Step 2: Substitute into the Difference Quotient Formula

Now, use the difference quotient formula:

$\frac{f(x+h) - f(x)}{h} = \frac{(7x + 7h + 1) - (7x + 1)}{h}$

Step 3: Simplify the Expression

Simplify the numerator:

$(7x + 7h + 1) - (7x + 1) = 7x + 7h + 1 - 7x - 1 = 7h$

Now, the difference quotient becomes:

$\frac{7h}{h}, \quad h \neq 0$

Step 4: Cancel $h$

Cancel out $h$ from the numerator and denominator:

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

Thus, the difference quotient is:

$7$

Final Answer:

The difference quotient of $f(x) = 7x + 1$ is $7$.

Would you like more details or have any questions?

Here are 5 related questions for further exploration:

1. What is the meaning of the difference quotient in calculus?
2. How would the result change if $f(x)$ was a quadratic function?
3. Can you find the derivative from the difference quotient?
4. How does the difference quotient relate to the concept of a secant line?
5. What happens if you apply the difference quotient to $f(x) = 0$?

Tip: The difference quotient is the foundation for understanding derivatives, which measure rates of change in calculus.