https://file.aiquickdraw.com/mathbot/file/1726956096616wc0kxdzq.jpg

We are tasked with finding the derivative of the function \( g(\theta) = \tan^2(5\theta) \).

### Step-by-step Solution:

1. **Rewrite the function**:
   \[
   g(\theta) = [\tan(5\theta)]^2
   \]
   
2. **Use the chain rule**:
   Let \( u = \tan(5\theta) \), then \( g(\theta) = u^2 \). The derivative of \( u^2 \) is \( 2u \cdot \frac{du}{d\theta} \).

3. **Differentiate the inner function**:
   \( u = \tan(5\theta) \), and its derivative is:
   \[
   \frac{d}{d\theta}[\tan(5\theta)] = 5 \sec^2(5\theta)
   \]

4. **Apply the chain rule**:
   Now, differentiate \( g(\theta) \):
   \[
   g'(\theta) = 2 \tan(5\theta) \cdot 5 \sec^2(5\theta)
   \]
   Simplifying:
   \[
   g'(\theta) = 10 \tan(5\theta) \sec^2(5\theta)
   \]

### Final Answer:
The derivative \( g'(\theta) = 10 \tan(5\theta) \sec^2(5\theta) \).

So, the correct answer is the third option in the image: **\( 10 \tan(5\theta) \sec^2(5\theta) \)**.

Let me know if you want any more details or have questions.

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Here are 5 related questions:

1. How do you apply the chain rule when differentiating composite functions?
2. What is the derivative of \( \sec(5\theta) \)?
3. How do you find the derivative of \( \tan^3(5\theta) \)?
4. What is the second derivative of \( g(\theta) = \tan^2(5\theta) \)?
5. How would you integrate \( \tan(5\theta) \sec^2(5\theta) \)?

**Tip**: When differentiating powers of trigonometric functions, remember to combine the chain rule and the power rule for precise results.

This problem demonstrates how to differentiate the function g(θ) = tan^2(5θ) by applying the chain rule and power rule. The solution walks through the step-by-step process, showing how to find g'(θ) = 10 * tan(5θ) * sec^2(5θ). Ideal for high school and college calculus students.

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