Definition The derivative of a function f at a number a, denoted by f9sad,
is
f9sad − lim
hl0
fsa 1 hd 2 fsad
h
if this limit exists.               Use this definition to find f ′(a) at the given number a.
f(x) = 

3x + 4
,    a = 4

To find the derivative of the function \( f(x) = 3x + 4 \) at \( a = 4 \), we will use the definition of the derivative:

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

### Step 1: Calculate \( f(a) \)
Given that \( f(x) = 3x + 4 \), for \( a = 4 \):

\[
f(4) = 3(4) + 4 = 12 + 4 = 16
\]

### Step 2: Calculate \( f(a + h) \)
Now, we calculate \( f(a + h) \), where \( a = 4 \):

\[
f(4 + h) = 3(4 + h) + 4 = 3(4) + 3h + 4 = 12 + 3h + 4 = 16 + 3h
\]

### Step 3: Apply the limit definition of the derivative
Now, plug \( f(4 + h) \) and \( f(4) \) into the definition of the derivative:

\[
f'(4) = \lim_{h \to 0} \frac{f(4 + h) - f(4)}{h} = \lim_{h \to 0} \frac{(16 + 3h) - 16}{h}
\]

Simplify the expression:

\[
f'(4) = \lim_{h \to 0} \frac{3h}{h} = \lim_{h \to 0} 3 = 3
\]

### Conclusion:
The derivative of \( f(x) = 3x + 4 \) at \( a = 4 \) is:

\[
f'(4) = 3
\]

Would you like further details or have any questions?

Here are 5 related questions to deepen understanding:
1. How would the derivative change if \( f(x) = 5x + 2 \)?
2. What is the geometric interpretation of the derivative at a point?
3. How does the derivative of a constant function behave?
4. Can you find the derivative of a quadratic function, like \( f(x) = x^2 \), using the definition of the derivative?
5. What does it mean if the derivative is zero at a point?

**Tip**: The derivative represents the slope of the tangent line to the graph of the function at a given point.

Definition: The derivative of a function f at a number a, denoted by f'(a), is f'(a) = lim (h -> 0) [(f(a + h) - f(a)) / h] if this limit exists. Use this definition to find f ′(a) at the given number a. f(x) = 3x + 4, a = 4.

Explore how to find the derivative of the linear function f(x) = 3x + 4 at the point a = 4 using the limit definition of derivatives. This detailed step-by-step solution covers fundamental calculus concepts and the application of the limit process. Ideal for students in Grades 11-12.

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