derivative of sec(theta)tan(theta) with respect to t

The derivative of \(\sec(\theta) \tan(\theta)\) with respect to \(t\) requires us to consider the chain rule since \(\theta\) is a function of \(t\).

### Step-by-Step Solution:
Given:
\[
f(\theta) = \sec(\theta) \tan(\theta)
\]
where \(\theta\) is a function of \(t\).

1. **Differentiate \(f(\theta)\) with respect to \(\theta\):**
   \[
   \frac{d}{d\theta} \left(\sec(\theta) \tan(\theta)\right)
   \]
   Using the product rule:
   \[
   \frac{d}{d\theta} (\sec(\theta) \tan(\theta)) = \sec(\theta) \frac{d}{d\theta} (\tan(\theta)) + \tan(\theta) \frac{d}{d\theta} (\sec(\theta))
   \]
   The derivatives of the trigonometric functions are:
   \[
   \frac{d}{d\theta} (\tan(\theta)) = \sec^2(\theta)
   \]
   \[
   \frac{d}{d\theta} (\sec(\theta)) = \sec(\theta) \tan(\theta)
   \]
   Substituting these into the product rule:
   \[
   \frac{d}{d\theta} (\sec(\theta) \tan(\theta)) = \sec(\theta) \sec^2(\theta) + \tan(\theta) \sec(\theta) \tan(\theta)
   \]
   Simplifying:
   \[
   = \sec^3(\theta) + \sec(\theta) \tan^2(\theta)
   \]
   \[
   = \sec(\theta)(\sec^2(\theta) + \tan^2(\theta))
   \]

2. **Apply the chain rule to find \(\frac{df}{dt}\):**
   \[
   \frac{df}{dt} = \frac{d}{d\theta} (\sec(\theta) \tan(\theta)) \cdot \frac{d\theta}{dt}
   \]
   Therefore:
   \[
   \frac{df}{dt} = \sec(\theta)(\sec^2(\theta) + \tan^2(\theta)) \cdot \frac{d\theta}{dt}
   \]

So, the derivative of \(\sec(\theta) \tan(\theta)\) with respect to \(t\) is:
\[
\frac{d}{dt} \left(\sec(\theta) \tan(\theta)\right) = \sec(\theta)(\sec^2(\theta) + \tan^2(\theta)) \cdot \frac{d\theta}{dt}
\]

Would you like more details or have any questions?

Here are 5 related questions:
1. How does the chain rule work for trigonometric functions?
2. What is the derivative of \(\tan(\theta)\) with respect to \(t\) if \(\theta = t^2\)?
3. Can you find the second derivative of \(\sec(\theta) \tan(\theta)\) with respect to \(t\)?
4. How would you differentiate a product of two trigonometric functions with respect to time?
5. What is the significance of the chain rule in calculus?

**Tip:** Always apply the chain rule when differentiating composite functions, especially when the inner function depends on the variable of differentiation.

Learn how to find the derivative of sec(theta) tan(theta) with respect to t using the chain rule and trigonometric derivatives. This problem involves advanced calculus concepts and is suitable for high school and college students.

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