To find the derivative \(\frac{dy}{dx}\) of the function \(Y = 2\cot(x) + \sec(x)\) at \(x = \frac{\pi}{6}\), follow these steps:

1. **Differentiate \(Y\) with respect to \(x\):**

   - The derivative of \(2\cot(x)\) with respect to \(x\) is:
     \[
     \frac{d}{dx}[2\cot(x)] = 2 \cdot \frac{d}{dx}[\cot(x)] = 2 \cdot (-\csc^2(x)) = -2\csc^2(x)
     \]
   - The derivative of \(\sec(x)\) with respect to \(x\) is:
     \[
     \frac{d}{dx}[\sec(x)] = \sec(x)\tan(x)
     \]

   Thus, the derivative \(\frac{dy}{dx}\) is:
   \[
   \frac{dy}{dx} = -2\csc^2(x) + \sec(x)\tan(x)
   \]

2. **Evaluate the derivative at \(x = \frac{\pi}{6}\):**

   - \(\csc\left(\frac{\pi}{6}\right)

Learn how to find the derivative of the function Y = 2cot(x) + sec(x) at x = π/6 with step-by-step instructions. This problem involves trigonometric differentiation and is suitable for advanced high school students and beyond.

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