https://file.aiquickdraw.com/mathbot/file/1726294245915oaahd2eu.jpg

We are given two functions in the image:

1. \( y = \sec(\theta) \)
2. \( u = \tan(\theta) \)

The task is to find \( \frac{d^2 y}{du^2} \), the second derivative of \( y \) with respect to \( u \).

### Step 1: Find \( \frac{dy}{du} \)

We use the chain rule to differentiate \( y \) with respect to \( u \). First, differentiate \( y \) with respect to \( \theta \), and then multiply it by \( \frac{d\theta}{du} \):

- \( \frac{dy}{d\theta} = \frac{d}{d\theta} \sec(\theta) = \sec(\theta) \tan(\theta) \)
- \( u = \tan(\theta) \) implies \( \frac{du}{d\theta} = \sec^2(\theta) \), so \( \frac{d\theta}{du} = \frac{1}{\sec^2(\theta)} \)

Thus:
\[
\frac{dy}{du} = \frac{dy}{d\theta} \cdot \frac{d\theta}{du} = \sec(\theta) \tan(\theta) \cdot \frac{1}{\sec^2(\theta)} = \frac{\tan(\theta)}{\sec(\theta)} = \sin(\theta)
\]

### Step 2: Find \( \frac{d^2 y}{du^2} \)

Now we differentiate \( \frac{dy}{du} = \sin(\theta) \) with respect to \( u \). Again, apply the chain rule:

- \( \frac{d}{d\theta} \sin(\theta) = \cos(\theta) \)
- Since \( \frac{du}{d\theta} = \sec^2(\theta) \), we multiply by \( \frac{d\theta}{du} = \frac{1}{\sec^2(\theta)} \)

Thus:
\[
\frac{d^2 y}{du^2} = \cos(\theta) \cdot \frac{1}{\sec^2(\theta)} = \frac{\cos(\theta)}{\sec^2(\theta)} = \frac{1}{\sec(\theta)} = \cos(\theta)
\]

Since \( u = \tan(\theta) \), we can express the result in terms of \( u \). Using the identity \( \sec^2(\theta) = 1 + \tan^2(\theta) \), we know that \( \cos(\theta) = \frac{1}{\sqrt{1 + u^2}} \).

Thus, the final answer is:
\[
\frac{d^2 y}{du^2} = \frac{1}{\sqrt{1 + u^2}}
\]

Would you like further clarification or details?

### Related questions:
1. What is the first derivative of \( \sec(\theta) \) with respect to \( \theta \)?
2. How is the chain rule applied in multivariable functions?
3. Can you provide other trigonometric identities used in calculus?
4. What is the physical interpretation of the second derivative in this context?
5. How would this calculation change if we used different trigonometric functions for \( y \) or \( u \)?

**Tip:** In chain rule problems, always carefully track which variable you're differentiating with respect to, especially when substituting multiple variables.

Find the second derivative of y = sec(θ) with respect to u = tan(θ).

Learn how to find the second derivative of sec(θ) with respect to tan(θ) using the chain rule in calculus. This step-by-step solution covers key trigonometric and calculus concepts, suitable for undergraduate students.

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