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.