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 a derivative, we apply the limit definition of the derivative:

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

In this case, the function is \( f(x) = 3x + 1 \) and \( a = 5 \). Let's calculate step-by-step:

### Step 1: Calculate \( f(a) \)
\[
f(a) = f(5) = 3(5) + 1 = 15 + 1 = 16
\]

### Step 2: Calculate \( f(a+h) \)
\[
f(a+h) = f(5+h) = 3(5+h) + 1 = 15 + 3h + 1 = 16 + 3h
\]

### Step 3: Apply the limit definition of the derivative
Now substitute these values into the derivative formula:

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

### Step 4: Simplify the expression
\[
f'(5) = \lim_{h \to 0} 3 = 3
\]

Thus, the derivative \( f'(5) = 3 \).

Would you like more details or have any questions about this? Here are some related questions for further understanding:

1. What is the general derivative of the function \( f(x) = 3x + 1 \)?
2. How is the derivative related to the slope of the function?
3. How can the limit definition of a derivative be applied to more complex functions?
4. What is the geometric interpretation of \( f'(a) \) for linear functions?
5. What happens to the derivative if we change \( a \) to a different value?

**Tip:** The derivative of a linear function \( f(x) = mx + b \) is always the slope \( m \).

Use this definition to find f'(a) at the given number a. f(x) = 3x + 1, a = 5

Learn how to use the limit definition of the derivative to find f'(5) for the linear function f(x) = 3x + 1. This step-by-step solution covers core calculus concepts such as limits, derivatives, and linear functions. Suitable for students in Grades 11-12 or introductory college-level calculus.

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