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

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 \( x \).