To analyze the question, we need to determine for which functions the chain rule is the appropriate method to differentiate.

Functions:

1. $y = \sin(3x^2)$
   The argument of the sine function, $3x^2$, is itself a function of $x$. Therefore, the chain rule is required to differentiate this composite function.

2. $y = e^x \cdot \tan(x)$
   Here, $y$ is the product of two functions of $x$, $e^x$ and $\tan(x)$. Differentiation will involve the product rule, not necessarily the chain rule.

3. $y = \frac{1}{8x^4 - 2x}$
   This is a rational function. To differentiate it, we can apply the quotient rule, but since the denominator $8x^4 - 2x$ is itself a composite function, the chain rule will be used in differentiating the denominator.

Conclusion:

The chain rule is applicable for:

• Function I ($y = \sin(3x^2)$)
• Function III ($y = \frac{1}{8x^4 - 2x}$)

Thus, the correct answer is D. I and III only.

Let me know if you'd like additional clarification or further details!

Relative Questions:

1. How does the chain rule differ from the product rule?
2. Can you provide examples of composite functions requiring the chain rule?
3. When would the quotient rule also require the chain rule?
4. How can trigonometric functions involve the chain rule in differentiation?
5. What are the steps to differentiate a rational function using the quotient rule and chain rule?

Tip: Always check whether a function is composite by identifying nested functions to decide if the chain rule is needed.