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The question asks to compute the derivative of \( \tan^3(x^2) \).

We can approach this by using the chain rule and power rule.

Given:
\[
f(x) = \tan^3(x^2)
\]

### Step 1: Use the chain rule
The function is \( \tan(x^2) \) raised to the power of 3. So we start by applying the power rule:

\[
\frac{d}{dx} \left[ \tan^3(x^2) \right] = 3 \cdot \tan^2(x^2) \cdot \frac{d}{dx} [ \tan(x^2) ]
\]

### Step 2: Differentiate \( \tan(x^2) \)
Now, we need to differentiate \( \tan(x^2) \) using the chain rule again. The derivative of \( \tan(u) \) is \( \sec^2(u) \), where \( u = x^2 \):

\[
\frac{d}{dx} [\tan(x^2)] = \sec^2(x^2) \cdot \frac{d}{dx}(x^2)
\]

The derivative of \( x^2 \) is \( 2x \), so:

\[
\frac{d}{dx} [\tan(x^2)] = 2x \sec^2(x^2)
\]

### Step 3: Combine the results
Now substitute this result back into the original expression:

\[
\frac{d}{dx} [\tan^3(x^2)] = 3 \cdot \tan^2(x^2) \cdot 2x \sec^2(x^2)
\]

Simplifying:

\[
= 6x \cdot \tan^2(x^2) \cdot \sec^2(x^2)
\]

### Final answer:
The correct option is **\( 6x \sec^2(x^2) \tan^2(x^2) \)**.

Would you like further details or have any questions?

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Here are 5 related questions to expand your understanding:

1. What is the derivative of \( \tan^n(x^2) \) for any integer \( n \)?
2. How does the chain rule apply when differentiating nested functions?
3. What are the derivatives of other trigonometric functions like \( \sin(x^2) \) or \( \cos(x^2) \)?
4. How would the result change if the power on \( \tan(x^2) \) was different?
5. Can you derive the second derivative of \( \tan^3(x^2) \)?

**Tip**: Always check if functions involve composition of functions, as this usually means applying the chain rule!

This solution explains how to compute the derivative of tan^3(x^2) using the chain rule and power rule. The derivative of the trigonometric function tan is involved, and the expression simplifies to 6x sec^2(x^2) tan^2(x^2). Suitable for students at the college calculus level.

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