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To find \( f'(x) \) for the function \( f(x) = (7x - 2x^3)(4 + \sqrt{x}) \), we can use the **product rule**. The product rule states that if \( f(x) = u(x)v(x) \), then

\[
f'(x) = u'(x)v(x) + u(x)v'(x).
\]

Let's define:
- \( u(x) = 7x - 2x^3 \)
- \( v(x) = 4 + \sqrt{x} \)

We need to find \( u'(x) \) and \( v'(x) \).

### Step 1: Differentiate \( u(x) \) and \( v(x) \)
1. **Differentiating \( u(x) = 7x - 2x^3 \):**
   \[
   u'(x) = 7 - 6x^2.
   \]

2. **Differentiating \( v(x) = 4 + \sqrt{x} \):**
   Rewrite \( \sqrt{x} \) as \( x^{1/2} \).
   \[
   v(x) = 4 + x^{1/2}.
   \]
   Then,
   \[
   v'(x) = \frac{1}{2}x^{-\frac{1}{2}} = \frac{1}{2\sqrt{x}}.
   \]

### Step 2: Apply the Product Rule
Now substitute into the product rule formula:

\[
f'(x) = u'(x)v(x) + u(x)v'(x).
\]

Substitute the values for \( u(x) \), \( u'(x) \), \( v(x) \), and \( v'(x) \):

\[
f'(x) = (7 - 6x^2)(4 + \sqrt{x}) + (7x - 2x^3) \cdot \frac{1}{2\sqrt{x}}.
\]

### Step 3: Simplify Each Term
1. **Expanding \( (7 - 6x^2)(4 + \sqrt{x}) \):**
   \[
   = 7 \cdot 4 + 7 \cdot \sqrt{x} - 6x^2 \cdot 4 - 6x^2 \cdot \sqrt{x}.
   \]
   Simplify each term:
   \[
   = 28 + 7\sqrt{x} - 24x^2 - 6x^{5/2}.
   \]

2. **Simplifying \( (7x - 2x^3) \cdot \frac{1}{2\sqrt{x}} \):**
   Distribute \(\frac{1}{2\sqrt{x}}\):
   \[
   = \frac{7x}{2\sqrt{x}} - \frac{2x^3}{2\sqrt{x}}.
   \]
   Simplify each term:
   \[
   = \frac{7\sqrt{x}}{2} - x^{5/2}.
   \]

### Step 4: Combine Terms
Now, add the terms from each part:

\[
f'(x) = 28 + 7\sqrt{x} - 24x^2 - 6x^{5/2} + \frac{7\sqrt{x}}{2} - x^{5/2}.
\]

This is the derivative \( f'(x) \) in its simplified form.

Would you like additional details on the steps or have any questions?

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**Relative Questions:**
1. How does the product rule work in calculus?
2. What is the process for differentiating power functions like \( x^{3/2} \)?
3. Why is simplifying square roots important in differentiation?
4. How do you interpret the result of \( f'(x) \) in terms of function behavior?
5. Can you explain the significance of each term in \( f'(x) \) after differentiation?

**Tip:** When using the product rule, double-check each derivative and simplify carefully to avoid errors.

Let f(x) = (7x - 2x^3)(4 + √x). Find f'(x).

Learn how to find the derivative of f(x) = (7x - 2x^3)(4 + √x) using the product rule in calculus. This step-by-step solution explains how to apply the product rule, differentiate powers of x, and simplify the result. Ideal for high school calculus students.

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