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📅 April 30, 2026 👤 scraface

Apr 30, 2026

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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 th...
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...
Set $ f'(u) = 0 \Rightarrow 1 - 81u^2 = 0 \Rightarrow u^2 = \frac{1}{8...
Recall $ u = e^{-0.5t} $, so:
e^{-0.5t} = \frac{1}{9} \Rightarrow -0.5t = \ln\left(\frac{1}{9}\right...

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