The derivative of the function $f(x) = \tan(3x)$ can be found using the chain rule.

Steps:

1. Derivative of $\tan(u)$: The derivative of $\tan(u)$ with respect to $u$ is: $\frac{d}{du}[\tan(u)] = \sec^2(u)$

2. Apply the Chain Rule: Let $u = 3x$. So, we apply the chain rule as follows: [ \frac{d}{dx}[\tan(3x)] = \sec^2(3x) \cdot \frac{d}{dx}[3x] ]

3. Differentiate $3x$: The derivative of $3x$ with respect to $x$ is 3: $\frac{d}{dx}[3x] = 3$

4. Final result: Now, combine the results: $f'(x) = 3 \sec^2(3x)$

Thus, the derivative of $f(x) = \tan(3x)$ is: $f'(x) = 3 \sec^2(3x)$

Would you like further explanation on any of the steps?

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Here are 5 related questions for practice:

1. What is the derivative of $\tan(5x)$?
2. How would you find the derivative of $\tan^2(x)$?
3. What is the derivative of $\sec(x)$?
4. How can you apply the product rule to $x^2 \cdot \tan(x)$?
5. How would you compute the derivative of $\cot(2x)$?

Tip: Remember, the chain rule is crucial when differentiating composite functions!