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To solve this problem, let's carefully analyze and apply the derivative rules as required. We’re given the function:

\[
f(x) = \frac{\sqrt{7x} \cdot (6 + x^2)}{\sqrt{2}}
\]

Since \(\sqrt{2}\) is a constant, we can factor it out, simplifying \(f(x)\) as:

\[
f(x) = \frac{1}{\sqrt{2}} \cdot \sqrt{7x} \cdot (6 + x^2)
\]

This form allows us to use the product rule to differentiate the expression with respect to \(x\).

Let:
\[
g(x) = \sqrt{7x} \quad \text{and} \quad h(x) = 6 + x^2
\]

Thus, \( f(x) = \frac{1}{\sqrt{2}} \cdot g(x) \cdot h(x) \).

### Step 1: Find \( g(x) \) and \( h(x) \) derivatives

1. **Derivative of \( g(x) = \sqrt{7x} \):**
   \[
   g(x) = (7x)^{\frac{1}{2}} \Rightarrow g'(x) = \frac{1}{2}(7x)^{-\frac{1}{2}} \cdot 7 = \frac{7}{2\sqrt{7x}}
   \]

2. **Derivative of \( h(x) = 6 + x^2 \):**
   \[
   h'(x) = 0 + 2x = 2x
   \]

### Step 2: Apply the Product Rule

The product rule states that:
\[
(fg)' = f'g + fg'
\]

Therefore:
\[
\frac{d}{dx} (g(x) \cdot h(x)) = g'(x) \cdot h(x) + g(x) \cdot h'(x)
\]

Substitute the derivatives found in Step 1:

\[
\frac{d}{dx} \left(\sqrt{7x} \cdot (6 + x^2)\right) = \frac{7}{2\sqrt{7x}} \cdot (6 + x^2) + \sqrt{7x} \cdot (2x)
\]

### Step 3: Substitute into the Final Expression

Now we have:
\[
f'(x) = \frac{1}{\sqrt{2}} \left( \frac{7}{2\sqrt{7x}} \cdot (6 + x^2) + \sqrt{7x} \cdot (2x) \right)
\]

This is the derivative of \(f(x)\) with respect to \(x\), as required.

Would you like a simplified version of this answer or further details on any steps?

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Here are some related questions that could deepen your understanding:

1. How does the product rule apply to functions with square roots?
2. What is the difference between the product rule and the quotient rule in calculus?
3. How would this derivative change if we added a constant to \( f(x) \)?
4. How would the answer differ if \( f(x) \) involved a cube root instead of a square root?
5. How can we verify the correctness of this derivative?

**Tip:** When simplifying derivatives, look for constants or terms that can factor out early. This can make calculations easier to manage.

Use the product rule to find the derivative with respect to the independent variable for f(x) = (sqrt(7x) * (6 + x^2)) / sqrt(2).

Step-by-step solution to differentiate f(x) = (sqrt(7x) * (6 + x^2)) / sqrt(2) using the product rule. This problem explores calculus concepts including the product rule and the derivative of square roots. Ideal for high school calculus students.

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