しあわせってなんだっけ

ある日、こんなツイートがRTされてきました。


人生ってなんだろう

 {
\begin{align}
\textit{Life} &= \displaystyle\int_{\textit{birth}}^{\textit{death}}\dfrac{\textit{happiness}}{\textit{time}}\Delta \textit{time} \\
 &= \textit{happiness} \displaystyle\int_{\textit{birth}}^{\textit{death}}\dfrac{1}{\textit{time}}\Delta \textit{time} \\
 &= \textit{happiness} \bigl[ \log \left| \textit{time} \right| \bigr]_{\textit{birth}}^{\textit{death}} \\
 &= \textit{happiness} \left( \log \left| \textit{death} \right| - \log \left| \textit{birth} \right| \right) \\
 &= \textit{happiness} \log \left| \dfrac{\textit{death}}{\textit{birth}} \right|
\end{align}
}

幸せってなんだろう

 {
\begin{align}
\textit{Life} &= \textit{happiness} \log \left| \dfrac{\textit{death}}{\textit{birth}} \right| \\
\textit{happiness} &= \dfrac{\textit{Life}}{\log \left| \frac{\textit{death}}{\textit{birth}} \right|} \\
 &= \dfrac{\textit{Life} \log e}{\log \left| \frac{\textit{death}}{\textit{birth}} \right|} \\
 &= \textit{Life}\log_{\left| \frac{\textit{death}}{\textit{birth}} \right|}e
\end{align}
}

おまけ:微分してみた

 {
\begin{align}
\textit{Life}(\textit{time}) &= \textit{happiness} \log (\textit{time}) \\
\textit{Life}'(\textit{time}) &= \dfrac{\textit{happiness}}{\textit{time}} \\
\textit{Life}''(\textit{time}) &= -\dfrac{\textit{happiness}}{\textit{time}^{2}} \\
\textit{Life}'''(\textit{time}) &= \dfrac{2\textit{happiness}}{\textit{time}^{3}} \\
&\vdots \\
\textit{Life}^{(n)}(\textit{time}) &= (-1)^{n-1}\dfrac{\left( n-1 \right)! \textit{happiness}}{\textit{time}^{n}} \quad(n \in \mathbb{Z}, n > 0) \\
\end{align}
}

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