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Home » Compute the third derivative \( p'''(x) \): - AdVision eCommerce

Compute the third derivative \( p'''(x) \): - AdVision eCommerce

📅 May 3, 2026 👤 scraface

May 03, 2026

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📌 Let \( f(x) = (\sec x + \csc x)^2 = \left( \frac{1}{\cos x} + \frac{1}{\sin x} \right)^2 \).

📌 f(x) = \left( \frac{\sin x + \cos x}{\sin x \cos x} \right)^2

📌 Let \( s = \sin x + \cos x \), \( p = \sin x \cos x \). Note:

📌 s^2 = \sin^2 x + \cos^2 x + 2\sin x \cos x = 1 + 2p \Rightarrow p = \frac{s^2 - 1}{2}

📌 f(x) = \left( \frac{s}{p} \right)^2 = \left( \frac{s}{(s^2 - 1)/2} \right)^2 = \left( \frac{2s}{s^2 - 1} \right)^2 = \frac{4s^2}{(s^2 - 1)^2}

📌 Now, for \( x \in (0, \frac{\pi}{2}) \), \( s = \sin x + \cos x = \sqrt{2} \sin\left(x + \frac{\pi}{4}\right) \), so \( s

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