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Part 1
Find and simplify the difference quotient StartFraction f left parenthesis x plus h right parenthesis minus f left parenthesis x right parenthesis Over h EndFraction
​, hnot equals0 for the given function.
​f(x)equals7x
Question content area bottom
Part 1
StartFraction f left parenthesis x plus h right parenthesis minus f left parenthesis x right parenthesis Over h EndFraction
equals
  
enter your response here ​(Simplify your​ answer.)

The difference quotient for a function \( f(x) \) is given by:

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

### Step 1: Given function
The given function is \( f(x) = 7x \).

### Step 2: Find \( f(x+h) \)
Substitute \( x+h \) into the function \( f(x) \):

\[
f(x+h) = 7(x+h) = 7x + 7h
\]

### Step 3: Compute the difference quotient
Now substitute \( f(x+h) \) and \( f(x) \) into the difference quotient formula:

\[
\frac{f(x+h) - f(x)}{h} = \frac{(7x + 7h) - 7x}{h}
\]

### Step 4: Simplify the expression
\[
\frac{7x + 7h - 7x}{h} = \frac{7h}{h}
\]

Since \( h \neq 0 \), the \( h \)'s cancel out:

\[
\frac{7h}{h} = 7
\]

### Final Answer:
The simplified difference quotient is:

\[
7
\]

Would you like more details, or do you have any questions?

Here are some related questions to explore:
1. What is the difference quotient for \( f(x) = x^2 \)?
2. How does the difference quotient relate to the derivative?
3. What happens to the difference quotient as \( h \to 0 \)?
4. Can you find the difference quotient for \( f(x) = 3x^3 \)?
5. How do you apply the difference quotient to functions with multiple terms?

**Tip:** The difference quotient is the foundation of finding the derivative of a function!

Find and simplify the difference quotient (f(x+h) - f(x)) / h, h ≠ 0 for the given function f(x) = 7x.

Learn how to simplify the difference quotient for the linear function f(x) = 7x. This step-by-step guide covers key calculus concepts like the difference quotient and linear functions, suitable for high school students in Grades 11-12.

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