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?

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!