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The problem asks us to find the first derivative of the function:
\[
f(x) = (x^2 + 1)(4x + 1).
\]

To solve this, we can use the **product rule** for derivatives, which states:
\[
(f \cdot g)' = f' \cdot g + f \cdot g',
\]
where \( f(x) = x^2 + 1 \) and \( g(x) = 4x + 1 \).

### Step-by-Step Solution

1. **Define** the two parts:
   - \( f(x) = x^2 + 1 \)
   - \( g(x) = 4x + 1 \)

2. **Compute** \( f'(x) \):
   \[
   f'(x) = \frac{d}{dx}(x^2 + 1) = 2x.
   \]

3. **Compute** \( g'(x) \):
   \[
   g'(x) = \frac{d}{dx}(4x + 1) = 4.
   \]

4. **Apply the Product Rule**:
   \[
   f'(x) = f'(x) \cdot g(x) + f(x) \cdot g'(x).
   \]
   Substituting the values, we get:
   \[
   f'(x) = (2x)(4x + 1) + (x^2 + 1)(4).
   \]

5. **Expand and Simplify**:
   \[
   f'(x) = 2x \cdot 4x + 2x \cdot 1 + 4 \cdot x^2 + 4 \cdot 1.
   \]
   This simplifies to:
   \[
   f'(x) = 8x^2 + 2x + 4x^2 + 4.
   \]

6. **Combine Like Terms**:
   \[
   f'(x) = 12x^2 + 2x + 4.
   \]

### Final Answer
The first derivative of \( f(x) \) is:
\[
f'(x) = 12x^2 + 2x + 4.
\]

Would you like more details on any part of this solution, or do you have any questions?

Here are some related questions for further practice:
1. How would you find the second derivative of \( f(x) \)?
2. What is the derivative of a function of the form \( (ax + b)(cx + d) \)?
3. How can the product rule be applied to three functions multiplied together?
4. What if \( f(x) = (x^2 + 1)(4x + 1)^2 \)? How would the differentiation change?
5. How can we apply the product and chain rules together?

**Tip:** Always double-check each step in differentiation to ensure no terms are missed, especially when applying product and chain rules.

Tentukan turunan pertama fungsi f(x) = (x^2 + 1)(4x + 1).

Learn how to find the first derivative of the function f(x) = (x^2 + 1)(4x + 1) using the product rule. This solution includes step-by-step instructions on applying differentiation concepts for students in Grades 11-12.

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