Differentiate.

*y* = sec(𝜃) tan(𝜃)

*y'* = 
 

To differentiate \( y = \sec(\theta) \tan(\theta) \), we will apply the product rule. The product rule states that if \( y = f(\theta) g(\theta) \), then:

\[
y' = f'(\theta) g(\theta) + f(\theta) g'(\theta)
\]

Here, \( f(\theta) = \sec(\theta) \) and \( g(\theta) = \tan(\theta) \).

### Step 1: Differentiate \( f(\theta) = \sec(\theta) \)
The derivative of \( \sec(\theta) \) is:
\[
f'(\theta) = \sec(\theta) \tan(\theta)
\]

### Step 2: Differentiate \( g(\theta) = \tan(\theta) \)
The derivative of \( \tan(\theta) \) is:
\[
g'(\theta) = \sec^2(\theta)
\]

### Step 3: Apply the product rule
Now, apply the product rule:
\[
y' = \sec(\theta) \tan(\theta) \cdot \sec^2(\theta) + \sec(\theta) \cdot \sec(\theta) \tan(\theta)
\]

Simplify the terms:
\[
y' = \sec^3(\theta) \tan(\theta) + \sec^2(\theta) \tan(\theta)
\]

Thus, the derivative is:
\[
y' = \sec^2(\theta) \tan(\theta)(\sec(\theta) + 1)
\]

Would you like further details on the steps or a more in-depth explanation?

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Here are 5 related questions to deepen your understanding:
1. What is the derivative of \( \sec(\theta) \) using its definition in terms of sine and cosine?
2. How would you differentiate \( \sin(\theta) \tan(\theta) \) instead?
3. Can you explain how the product rule works for three functions?
4. What are the second derivatives of trigonometric functions like \( \sec(\theta) \) and \( \tan(\theta) \)?
5. How can you apply implicit differentiation to equations involving trigonometric functions?

**Tip:** Always check if trigonometric identities can simplify the function before differentiation.

Learn how to differentiate the function y = sec(𝜃) tan(𝜃) using the product rule. This solution covers key concepts in calculus and trigonometry, including the derivatives of sec(𝜃) and tan(𝜃). Suitable for students in Grades 11-12.

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