To find $f'(a)$ using the definition of the derivative, we use the limit definition:

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

Given:

• $f(x) = 3x + 1$
• $a = 5$

We can now follow the steps to compute the derivative.

Step 1: Evaluate $f(a + h)$

$f(a + h) = f(5 + h) = 3(5 + h) + 1 = 15 + 3h + 1 = 16 + 3h$

Step 2: Evaluate $f(a)$

$f(a) = f(5) = 3(5) + 1 = 15 + 1 = 16$

Step 3: Substitute into the definition of the derivative

$f'(5) = \lim_{h \to 0} \frac{(16 + 3h) - 16}{h} = \lim_{h \to 0} \frac{3h}{h}$

Step 4: Simplify and take the limit

$f'(5) = \lim_{h \to 0} 3 = 3$

So, $f'(5) = 3$.

Would you like further details or have any questions about this?

Here are 5 questions related to this:

1. What is the general form of the derivative of a linear function?
2. How would the derivative change if $f(x)$ was quadratic instead of linear?
3. Can the derivative of $f(x) = 3x + 1$ change for different values of $a$?
4. What does the derivative tell us about the slope of the function at a given point?
5. How can you interpret $f'(a)$ in real-world applications?

Tip: The derivative of a linear function $f(x) = mx + b$ is always the constant $m$, representing the slope of the line.