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super mario galaxy 2 switch - Imagemakers

📅 April 29, 2026 👤 scraface

Apr 29, 2026

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📌 To find the maximum of $ P'(t) $, we define:

📌 f(t) = \frac{e^{-0.5t}}{(1 + 9e^{-0.5t})^2}.

📌 Let $ u = e^{-0.5t} $, so $ f(t) = \frac{u}{(1 + 9u)^2} $. Now find the maximum of this function with respect to $ u > 0 $.

📌 f(u) = \frac{u}{(1 + 9u)^2}.

📌 Differentiate using the quotient rule:

📌 f'(u) = \frac{(1 + 9u)^2 \cdot 1 - u \cdot 2(1 + 9u)(9)}{(1 + 9u)^4}.

📌 Simplify numerator:

📌 f'(u) = \frac{(1 + 9u)(1 - 9u)}{(1 + 9u)^4} = \frac{1 - 81u^2}{(1 + 9u)^3}.

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