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.