The expression $\frac{f(x + h) - f(x)}{h}$ is a common difference quotient used in calculus to compute the derivative of a function.

Given $f(x) = 6x - 6$, let's compute the difference quotient step by step:

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

Substitute $x + h$ into the function: $f(x + h) = 6(x + h) - 6 = 6x + 6h - 6$

Step 2: Compute $f(x + h) - f(x)$

Now subtract $f(x) = 6x - 6$ from $f(x + h)$: $f(x + h) - f(x) = (6x + 6h - 6) - (6x - 6) = 6h$

Step 3: Divide by $h$

Now divide the result by $h$: $\frac{f(x + h) - f(x)}{h} = \frac{6h}{h} = 6$

Thus, the difference quotient $\frac{f(x + h) - f(x)}{h}$ simplifies to 6.

Would you like more details or have any questions?

Related questions:

1. What is the significance of the difference quotient?
2. How is the difference quotient related to the derivative of a function?
3. How would this process change if $f(x)$ were a quadratic function?
4. What happens to the difference quotient as $h \to 0$?
5. How can the difference quotient help in understanding linear functions?

Tip:

For linear functions like $f(x) = 6x - 6$, the difference quotient always simplifies to the constant slope of the line.