Use this definition to find *f* ′(*a*) at the given number *a*.

*f*(*x*) = 

3*x* + 1

,    *a* = 5

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.

Use this definition to find f′(a) at the given number a.

f(x) = 3x + 1, a = 5

Learn how to find the derivative f'(5) for the linear function f(x) = 3x + 1 using the limit definition of the derivative. This problem walks through the step-by-step process of solving for f'(a) and simplifies the solution for high school calculus students.

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