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To solve the problem, we need to find the derivative of the function \( f(x) = \sqrt{6x^2 + 6x^4 + 8x} \). Here is a step-by-step approach:

### Step 1: Rewrite the function in a more derivative-friendly form
The given function is:
\[
f(x) = \sqrt{6x^2 + 6x^4 + 8x}.
\]
We can rewrite the square root as a power of 1/2:
\[
f(x) = \left(6x^2 + 6x^4 + 8x\right)^{1/2}.
\]

### Step 2: Apply the chain rule
To differentiate \( f(x) \), we use the chain rule. The chain rule states:
\[
\frac{d}{dx} \left[ g(x)^n \right] = n \cdot g(x)^{n-1} \cdot g'(x),
\]
where \( g(x) = 6x^2 + 6x^4 + 8x \) and \( n = 1/2 \).

Thus, the derivative of \( f(x) \) is:
\[
f'(x) = \frac{1}{2} \cdot \left(6x^2 + 6x^4 + 8x\right)^{-1/2} \cdot \frac{d}{dx} \left(6x^2 + 6x^4 + 8x\right).
\]

### Step 3: Differentiate the inside function
Next, we differentiate the expression inside the parentheses:
\[
\frac{d}{dx} \left(6x^2 + 6x^4 + 8x\right) = 12x + 24x^3 + 8.
\]

### Step 4: Combine everything
Now substitute the derivative of the inside function back into the expression for \( f'(x) \):
\[
f'(x) = \frac{1}{2} \cdot \left(6x^2 + 6x^4 + 8x\right)^{-1/2} \cdot \left(12x + 24x^3 + 8\right).
\]

### Final Expression:
Thus, the derivative of the function is:
\[
f'(x) = \frac{12x + 24x^3 + 8}{2\sqrt{6x^2 + 6x^4 + 8x}}.
\]

Would you like more details on any of these steps? Here are 5 related questions:

1. How does the chain rule apply to this specific problem?
2. What would happen if the exponent on the entire function were different, such as \( f(x) = (6x^2 + 6x^4 + 8x)^n \)?
3. How do we simplify expressions that involve square roots and polynomials?
4. Can the function \( 6x^2 + 6x^4 + 8x \) be factored to simplify the derivative further?
5. What is the importance of the power rule in calculus for problems like this?

### Tip:
When differentiating a function involving a square root, always rewrite the square root as a fractional exponent to make applying the chain rule easier.

Find the derivative of the function: f(x) = sqrt(6x^2 + 6x^4 + 8x).

Learn how to find the derivative of the function f(x) = sqrt(6x^2 + 6x^4 + 8x) using the chain rule. This step-by-step guide explains the application of calculus concepts, such as differentiation and the chain rule. Suitable for advanced high school or university-level students.

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