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

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

Given $f(x) = 3x^2 - 5x - 9$, let's compute the difference quotient step by step.

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

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

$f(x+h) = 3(x+h)^2 - 5(x+h) - 9$

First, expand $(x+h)^2$:

$(x+h)^2 = x^2 + 2xh + h^2$

Now substitute back:

$f(x+h) = 3(x^2 + 2xh + h^2) - 5(x+h) - 9$

Expand the terms:

$f(x+h) = 3x^2 + 6xh + 3h^2 - 5x - 5h - 9$

Step 2: Subtract $f(x)$

Now subtract $f(x) = 3x^2 - 5x - 9$ from $f(x+h)$:

$f(x+h) - f(x) = (3x^2 + 6xh + 3h^2 - 5x - 5h - 9) - (3x^2 - 5x - 9)$

Simplify the expression:

$f(x+h) - f(x) = 6xh + 3h^2 - 5h$

Step 3: Divide by $h$

Now, divide by $h$:

$\frac{f(x+h) - f(x)}{h} = \frac{6xh + 3h^2 - 5h}{h}$

Simplify:

$\frac{f(x+h) - f(x)}{h} = 6x + 3h - 5$

Thus, the difference quotient is:

$6x + 3h - 5$

Would you like further details or have any questions?

Related Questions:

1. How does the difference quotient relate to the derivative?
2. What is the significance of $h$ in the difference quotient formula?
3. How would the difference quotient change if $f(x)$ were a cubic function?
4. Can we simplify the difference quotient when $h = 0$?
5. How can we use the difference quotient to find the slope of the tangent line?

Tip:

The difference quotient is a key concept for understanding the derivative, as $h$ approaches 0.