26. Find the derivative of the function, $$ f(x)=3 x-8 $$, using the definitio

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To find the derivative of the function $$ f(x) = 3x - 8 $$ using the **definition of the derivative**, we'll follow these steps:

### Definition of the Derivative

The derivative of a function $$ f(x) $$ at a point $$ x $$ is defined as:

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

### Step-by-Step Calculation

1. **Compute $$ f(x + h) $$:**

$$
   f(x + h) = 3(x + h) - 8 = 3x + 3h - 8
   $$

2. **Find the Difference $$ f(x + h) - f(x) $$:**

$$
   f(x + h) - f(x) = (3x + 3h - 8) - (3x - 8) = 3x + 3h - 8 - 3x + 8 = 3h
   $$

3. **Form the Difference Quotient:**

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

4. **Take the Limit as $$ h \to 0 $$:**

$$
   f'(x) = \lim_{h \to 0} 3 = 3
   $$

### Conclusion

The derivative of the function $$ f(x) = 3x - 8 $$ is:

$$
f'(x) = 3
$$

### Interpretation

- **Constant Derivative:** Since $$ f(x) $$ is a linear function (a straight line), its slope is constant. The derivative $$ f'(x) = 3 $$ represents this constant slope.
- **Geometric Meaning:** The slope of the line $$ f(x) = 3x - 8 $$ is 3, which means for every unit increase in $$ x $$, $$ f(x) $$ increases by 3 units.

This result aligns with our expectation for linear functions, where the derivative is simply the coefficient of $$ x $$.
The derivative of $$ f(x) = 3x - 8 $$ is $$ f'(x) = 3 $$.

Find the derivative of the function, , using the definitio

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