To solve the problem, we need to determine the derivative $f'(x)$ of the given function:

$f(x) = 6^{4x^2 - 4x + 8}.$

Step-by-Step Solution

1. Recall the derivative of exponential functions: If $y = a^{g(x)}$, then $y' = a^{g(x)} \ln(a) \cdot g'(x).$

   Here, $a = 6$ and $g(x) = 4x^2 - 4x + 8$.

2. Apply the formula: The derivative of $f(x)$ is: $f'(x) = 6^{4x^2 - 4x + 8} \ln(6) \cdot \frac{d}{dx}(4x^2 - 4x + 8).$

3. Differentiate $g(x) = 4x^2 - 4x + 8$: Using standard rules of differentiation: $g'(x) = \frac{d}{dx}(4x^2) - \frac{d}{dx}(4x) + \frac{d}{dx}(8),$ which simplifies to: $g'(x) = 8x - 4.$

4. Combine results: Substitute $g'(x) = 8x - 4$ back into the formula for $f'(x)$: $f'(x) = 6^{4x^2 - 4x + 8} \ln(6) (8x - 4).$

5. Match with the options: The correct answer is Option 3: $f'(x) = 6^{4x^2 - 4x + 8} \ln(6) (8x - 4).$

Final Answer:

$\text{Odgovor je: } 3.$

Do you want further clarification on any step? Here are 5 related questions you can explore:

1. Why is the chain rule important in differentiating exponential functions?
2. How does the natural logarithm $\ln(a)$ appear in the derivative of exponential functions?
3. Can you derive the derivative of another similar function, such as $3^{x^2 + x}$?
4. What is the role of the constant $8x - 4$ in the derivative?
5. How does this derivative help in analyzing the behavior of the function $f(x)$?

Tip: Always break down functions into inner and outer components to apply the chain