To find the difference quotient $\frac{f(x+h) - f(x)}{h}$, we'll follow these steps:

Step 1: Define $f(x)$

Given: $f(x) = x^2 - 10x$

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

We need to substitute $x+h$ into the function $f(x)$: $f(x+h) = (x+h)^2 - 10(x+h)$

Expanding: $f(x+h) = (x^2 + 2xh + h^2) - 10(x + h)$ $f(x+h) = x^2 + 2xh + h^2 - 10x - 10h$

Step 3: Compute $f(x+h) - f(x)$

Now, subtract $f(x)$ from $f(x+h)$: $f(x+h) - f(x) = (x^2 + 2xh + h^2 - 10x - 10h) - (x^2 - 10x)$

Simplify: $f(x+h) - f(x) = x^2 + 2xh + h^2 - 10x - 10h - x^2 + 10x$ $f(x+h) - f(x) = 2xh + h^2 - 10h$

Step 4: Divide by $h$

Now, divide the result by $h$: $\frac{f(x+h) - f(x)}{h} = \frac{2xh + h^2 - 10h}{h}$

Simplify: $\frac{f(x+h) - f(x)}{h} = 2x + h - 10$

Final Answer:

The difference quotient is: $\frac{f(x+h) - f(x)}{h} = 2x + h - 10$

Do you have any questions, or would you like more details?

Here are 5 related questions to expand on this concept:

1. How does the difference quotient relate to the derivative of a function?
2. What happens to the difference quotient as $h$ approaches 0?
3. Can you compute the derivative of $f(x) = x^2 - 10x$ using limits?
4. How would the difference quotient change if the function were $f(x) = x^3 - 10x$?
5. What is the geometric interpretation of the difference quotient?

Tip: The difference quotient is a key concept for understanding the derivative, which measures the rate of change of a function.