Y=sec($$ \frac{\pi}{4})x,\th\exponentialE n\frac{\text{ dy}}{\differentialD x}= $$

Let's start by calculating the derivative of the given function $ Y = \sec\left(\frac{\pi}{4}x\right) $.

The derivative of the secant function can be found using the following rule:

\[
\frac{d}{dx}[\sec(u)] = \sec(u) \tan(u) \frac{du}{dx}
\]

### Steps:

1. **Identify the function and its argument**:
   - $ u(x) = \frac{\pi}{4}x $
   - So, $ Y = \sec\left(\frac{\pi}{4}x\right) $.

2. **Find the derivative of the argument** $ u(x) = \frac{\pi}{4}x $:
   \[
   \frac{du}{dx} = \frac{\pi}{4}
   \]

3. **Apply the chain rule**:
   Using the derivative of the secant function:
   \[
   \frac{dy}{dx} = \sec\left(\frac{\pi}{4}x\right) \tan\left(\frac{\pi}{4}x\right) \cdot \frac{\pi}{4}
   \]

Thus, the derivative of $ Y = \sec\left(\frac{\pi}{4}x\right) $ is:

\[
\frac{dy}{dx} = \frac{\pi}{4} \sec\left(\frac{\pi}{4}x\right) \tan\left(\frac{\pi}{4}x\right)
\]

Would you like more details, or do you have any questions?

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Here are 5 related questions:

1. How do we differentiate the tangent function?
2. What is the derivative of $ \sin(x) $ and $ \cos(x) $?
3. How do trigonometric identities help in simplifying derivatives?
4. What is the chain rule in differentiation?
5. How do you apply the product rule to more complex trigonometric functions?

**Tip**: When working with trigonometric functions, knowing key identities like $ \sec(x) = \frac{1}{\cos(x)} $ can simplify many problems.

Y = sec(\frac{\pi}{4}x), find \frac{dy}{dx}

Learn how to differentiate the function Y = sec(\frac{\pi}{4}x) using the chain rule and the derivative of the secant function. This solution breaks down the steps for applying key calculus rules, suitable for college-level calculus students.

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