Determine the difference quotient f(x+h)-f(x)/h. f(x)=x^2-10x

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.

Learn how to calculate the difference quotient for the function f(x) = x^2 - 10x. This step-by-step solution explains key concepts such as the difference quotient, polynomial expansion, and simplification. Suitable for students in Grades 9-12.

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