Use the definition of a derivative to find fprime f(x)= ((2x)/7)+1

To find the derivative of \( f(x) = \frac{2x}{7} + 1 \) using the definition of a derivative, we start with the formula:

\[
f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}
\]

### Step 1: Substitute \( f(x) \) and \( f(x + h) \)
The given function is 
\[
f(x) = \frac{2x}{7} + 1
\]
First, find \( f(x + h) \):
\[
f(x + h) = \frac{2(x + h)}{7} + 1 = \frac{2x + 2h}{7} + 1
\]

### Step 2: Calculate \( f(x + h) - f(x) \)
\[
f(x + h) - f(x) = \left( \frac{2x + 2h}{7} + 1 \right) - \left( \frac{2x}{7} + 1 \right)
\]
Simplify this expression:
\[
f(x + h) - f(x) = \frac{2x}{7} + \frac{2h}{7} + 1 - \frac{2x}{7} - 1
\]
\[
f(x + h) - f(x) = \frac{2h}{7}
\]

### Step 3: Divide by \( h \) and take the limit
\[
\frac{f(x + h) - f(x)}{h} = \frac{\frac{2h}{7}}{h} = \frac{2}{7}
\]

Taking the limit as \( h \to 0 \):
\[
f'(x) = \lim_{h \to 0} \frac{2}{7} = \frac{2}{7}
\]

### Conclusion
The derivative of \( f(x) = \frac{2x}{7} + 1 \) is 
\[
f'(x) = \frac{2}{7}
\]

Would you like any further details or have any questions?

### 5 Relative Questions
1. How do you use the definition of a derivative for a more complex function?
2. What is the geometric interpretation of the derivative \( f'(x) = \frac{2}{7} \)?
3. How would you find the derivative of a quadratic function using the definition?
4. What happens if the function has a different slope at various points?
5. How does the definition of a derivative connect to instantaneous rates of change?

### Tip
The definition of a derivative gives the slope of the tangent line to the function at any point, and it's crucial for understanding how functions change locally.

This article provides a step-by-step guide on how to use the definition of a derivative to differentiate the linear function f(x) = (2x/7) + 1. Learn how limits are applied and how to simplify the process. Suitable for high school students studying calculus.

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