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To find the first derivative of the function \( f(x) = (x^2 + 2x)(3x - 1) \), we can apply the **product rule** for differentiation. The product rule states that for two functions \( u(x) \) and \( v(x) \), the derivative is given by:

\[
\frac{d}{dx} [u(x) \cdot v(x)] = u'(x) \cdot v(x) + u(x) \cdot v'(x)
\]

Here:
- \( u(x) = x^2 + 2x \)
- \( v(x) = 3x - 1 \)

### Step 1: Differentiate each part
- \( u'(x) = \frac{d}{dx}(x^2 + 2x) = 2x + 2 \)
- \( v'(x) = \frac{d}{dx}(3x - 1) = 3 \)

### Step 2: Apply the product rule
Now, using the product rule:

\[
f'(x) = u'(x) \cdot v(x) + u(x) \cdot v'(x)
\]
\[
f'(x) = (2x + 2) \cdot (3x - 1) + (x^2 + 2x) \cdot 3
\]

### Step 3: Simplify the expression
Expand the terms:
1. \( (2x + 2)(3x - 1) = 6x^2 - 2x + 6x - 2 = 6x^2 + 4x - 2 \)
2. \( 3(x^2 + 2x) = 3x^2 + 6x \)

Now combine both parts:
\[
f'(x) = (6x^2 + 4x - 2) + (3x^2 + 6x)
\]
\[
f'(x) = 6x^2 + 3x^2 + 4x + 6x - 2
\]
\[
f'(x) = 9x^2 + 10x - 2
\]

Thus, the first derivative of the function is:
\[
f'(x) = 9x^2 + 10x - 2
\]

Would you like further clarification or a detailed step on any part?

Here are 5 related questions to deepen your understanding:
1. How can you apply the chain rule in more complex functions?
2. What happens when you differentiate a quotient of two functions?
3. Can you find the second derivative of this function?
4. How can you apply this differentiation to real-world rate-of-change problems?
5. What are critical points and how do they relate to the first derivative?

**Tip:** Always simplify the expression after using product or chain rule to avoid unnecessary complexity.

Tentukan turunan pertama fungsi f(x) = (x^2 + 2x)(3x - 1)

Learn how to find the first derivative of the function f(x) = (x^2 + 2x)(3x - 1) using the product rule. This step-by-step guide covers key calculus concepts such as differentiation and the product rule, suitable for Grades 10-12 students.

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