The expression \( 3 \tan^{\prime4} 3x \) appears to involve a derivative and the fourth power of the tangent function. I'll interpret this as finding the derivative of \(3 \tan^4(3x)\).

To clarify, we will differentiate \(3 \tan^4(3x)\) with respect to \(x\).

### Step-by-Step Solution
1. **Rewrite the expression:**
   \[
   f(x) = 3 \tan^4(3x)
   \]

2. **Apply the constant factor rule:**
   Since 3 is a constant factor, we can pull it out of the derivative:
   \[
   f'(x) = 3 \cdot \frac{d}{dx} \left( \tan^4(3x) \right)
   \]

3. **Differentiate using the chain rule:**
   To differentiate \(\tan^4(3x)\), we apply the chain rule.

   Let \( u = \tan(3x) \). Then \( \tan^4(3x) = u^4 \), so we differentiate \(u^4\) with respect to \(u\) and then \(u\) with respect to \(x\).

   \[
   \frac{d}{dx} \left( \tan^4(3x) \right) = 4 \tan^3(3x) \cdot \frac{d}{dx} (\tan(3x))
   \]

4. **Differentiate \(\tan(3x)\) with respect to \(x\):**
   \[
   \frac{d}{dx} (\tan(3x)) = 3 \sec^2(3x)
   \]

5. **Combine the derivatives:**
   Substituting back, we get:
   \[
   f'(x) = 3 \cdot \left( 4 \tan^3(3x) \cdot 3 \sec^2(3x) \right)
   \]

6. **Simplify:**
   \[
   f'(x) = 3 \cdot 12 \tan^3(3x) \sec^2(3x) = 36 \tan^3(3x) \sec^2(3x)
   \]

### Final Answer
\[
f'(x) = 36 \tan^3(3x) \sec^2(3x)
\]

Would you like further details or have any questions about the solution steps?

---

Here are some additional questions that might interest you:
1. How would you differentiate \(\tan^n(kx)\) for a general \(n\) and constant \(k\)?
2. What is the derivative of \(\sec(x)\) and how can it be derived?
3. Can we integrate \( \tan^3(x) \sec^2(x) \) and find a closed-form solution?
4. How does the chain rule differ when applied to trigonometric functions like \(\sin\), \(\cos\), and \(\tan\)?
5. What is the significance of \(\sec(x)\) in trigonometric identities and calculus?

**Tip:** When differentiating powers of trigonometric functions, remember to apply both the power rule and chain rule in sequence to ensure accuracy.

Explore the step-by-step solution for differentiating the expression 3tan⁴(3x). This calculus problem applies the chain rule and power rule to trigonometric functions, perfect for advanced high school or college calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation