Compute ,f’’(x) if ,f(x) = (2x+ 3)^2(x^3+ 3)

We are tasked with finding the second derivative \( f''(x) \) of the function:

\[
f(x) = (2x + 3)^2 (x^3 + 3)
\]

To do this, we will first compute the first derivative \( f'(x) \), and then differentiate again to obtain \( f''(x) \).

### Step 1: First Derivative \( f'(x) \)

We use the product rule to differentiate \( f(x) \), which is the product of two functions. The product rule states:

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

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

Now, we differentiate \( u(x) \) and \( v(x) \):

#### 1. Derivative of \( u(x) = (2x + 3)^2 \):
Using the chain rule:

\[
u'(x) = 2(2x + 3) \cdot \frac{d}{dx}(2x + 3) = 2(2x + 3) \cdot 2 = 4(2x + 3)
\]

#### 2. Derivative of \( v(x) = (x^3 + 3) \):
\[
v'(x) = 3x^2
\]

Now apply the product rule:

\[
f'(x) = u'(x) v(x) + u(x) v'(x)
\]
Substitute \( u'(x) \), \( u(x) \), \( v'(x) \), and \( v(x) \):

\[
f'(x) = 4(2x + 3)(x^3 + 3) + (2x + 3)^2(3x^2)
\]

### Step 2: Simplify the first derivative \( f'(x) \)

We will simplify each term of \( f'(x) \).

1. First term: \( 4(2x + 3)(x^3 + 3) \)

Distribute:

\[
4(2x + 3)(x^3 + 3) = 4[(2x)(x^3) + (2x)(3) + (3)(x^3) + (3)(3)]
= 4(2x^4 + 6x + 3x^3 + 9) = 8x^4 + 12x^3 + 24x + 36
\]

2. Second term: \( (2x + 3)^2(3x^2) \)

Expand \( (2x + 3)^2 \):

\[
(2x + 3)^2 = (2x + 3)(2x + 3) = 4x^2 + 12x + 9
\]

Now multiply by \( 3x^2 \):

\[
(4x^2 + 12x + 9)(3x^2) = 12x^4 + 36x^3 + 27x^2
\]

Thus, the first derivative is:

\[
f'(x) = (8x^4 + 12x^3 + 24x + 36) + (12x^4 + 36x^3 + 27x^2)
\]

Combine like terms:

\[
f'(x) = (8x^4 + 12x^4) + (12x^3 + 36x^3) + 27x^2 + 24x + 36
\]
\[
f'(x) = 20x^4 + 48x^3 + 27x^2 + 24x + 36
\]

### Step 3: Second Derivative \( f''(x) \)

Now we differentiate \( f'(x) \) to find \( f''(x) \):

\[
f'(x) = 20x^4 + 48x^3 + 27x^2 + 24x + 36
\]

Differentiate term by term:

\[
f''(x) = \frac{d}{dx}[20x^4] + \frac{d}{dx}[48x^3] + \frac{d}{dx}[27x^2] + \frac{d}{dx}[24x] + \frac{d}{dx}[36]
\]
\[
f''(x) = 80x^3 + 144x^2 + 54x + 24
\]

Thus, the second derivative is:

\[
f''(x) = 80x^3 + 144x^2 + 54x + 24
\]

### Final Answer:
\[
f''(x) = 80x^3 + 144x^2 + 54x + 24
\]

---

Would you like further clarification or details on any of these steps?

Here are 5 related questions:
1. What is the third derivative \( f'''(x) \) of this function?
2. How do we apply the product rule when we have more than two factors?
3. Can we use logarithmic differentiation to simplify problems like this?
4. How would the derivative change if the exponents were fractional or negative?
5. How do implicit differentiation and chain rule combine in complex problems?

**Tip:** When differentiating products, always clearly identify each function before applying the product rule!

Compute f''(x) if f(x) = (2x + 3)^2(x^3 + 3)

Learn how to compute the second derivative f''(x) for the function f(x) = (2x + 3)^2(x^3 + 3) using the product and chain rules. This detailed step-by-step solution simplifies complex calculus concepts, including differentiation of products and polynomials, suitable for advanced high school and undergraduate students.

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