Find the derivative of the function f
 given by

f(x)=tan(3x)

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!

Find the derivative of the function f given by f(x) = tan(3x)

Learn how to find the derivative of the function f(x) = tan(3x) using the chain rule. This step-by-step solution covers differentiation of trigonometric functions and applies the chain rule, suitable for students in Grades 11-12.

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