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Home » Factor \( (1 + t^3) \) from numerator: - ClickBalance

Factor \( (1 + t^3) \) from numerator: - ClickBalance

📅 May 17, 2026 👤 scraface

May 17, 2026

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To find the time at which the rate of change is maximized, we compute...
Let’s denote the denominator as $ D(t) = 1 + 9e^{-0.5t} $, so:
P(t) = \frac{1000}{D(t)} \Rightarrow P'(t) = -1000 \cdot \frac{D'(t)}{...
Now compute $ D'(t) $:
D(t) = 1 + 9e^{-0.5t} \Rightarrow D'(t) = 9 \cdot (-0.5)e^{-0.5t} = -4...
P'(t) = -1000 \cdot \frac{-4.5e^{-0.5t}}{(1 + 9e^{-0.5t})^2} = \frac{4...
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 th...
f(u) = \frac{u}{(1 + 9u)^2}.

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