\(\cosh{\left(x\right)}\)の微分
\begin{eqnarray}
\frac{\mathrm{d}}{\mathrm{d}x}\cosh{\left(x\right)}
&=& \frac{\mathrm{d}}{\mathrm{d}x}\frac{e^x+e^{-x}}{2}
\\&=& \frac{1}{2}\left(\frac{\mathrm{d}}{\mathrm{d}x}e^x+\frac{\mathrm{d}}{\mathrm{d}x}e^{-x}\right)
\\&&\;\ldots\;\frac{\mathrm{d}}{\mathrm{d}x}Cf(x)=C\frac{\mathrm{d}}{\mathrm{d}x}f(x)
\\&&\;\ldots\;\frac{\mathrm{d}}{\mathrm{d}x}\left(f(x)+g(x)\right)=\frac{\mathrm{d}}{\mathrm{d}x}f(x)+\frac{\mathrm{d}}{\mathrm{d}x}g(x)
\\&=& \frac{1}{2}\left(\frac{\mathrm{d}}{\mathrm{d}x}e^x+\frac{\mathrm{d}}{\mathrm{d}x}e^{-x}\right)
\\&=& \frac{1}{2}\left\{e^x+(-1)e^{-x}\right\}
\\&=& \frac{1}{2}\left(e^x-e^{-x}\right)
\\&=& \sinh{\left(x\right)}
\end{eqnarray}
\(\sinh{\left(x\right)}\)の微分
\begin{eqnarray}
\frac{\mathrm{d}}{\mathrm{d}x}\sinh{\left(x\right)}
&=& \frac{\mathrm{d}}{\mathrm{d}x}\frac{e^x-e^{-x}}{2}
\\&=& \frac{1}{2}\left(\frac{\mathrm{d}}{\mathrm{d}x}e^x-\frac{\mathrm{d}}{\mathrm{d}x}e^{-x}\right)
\\&&\;\ldots\;\frac{\mathrm{d}}{\mathrm{d}x}Cf(x)=C\frac{\mathrm{d}}{\mathrm{d}x}f(x)
\\&&\;\ldots\;\frac{\mathrm{d}}{\mathrm{d}x}\left(f(x)+g(x)\right)=\frac{\mathrm{d}}{\mathrm{d}x}f(x)+\frac{\mathrm{d}}{\mathrm{d}x}g(x)
\\&=& \frac{1}{2}\left(\frac{\mathrm{d}}{\mathrm{d}x}e^x-\frac{\mathrm{d}}{\mathrm{d}x}e^{-x}\right)
\\&=& \frac{1}{2}\left\{e^x-(-1)e^{-x}\right\}
\\&=& \frac{1}{2}\left(e^x+e^{-x}\right)
\\&=& \cosh{\left(x\right)}
\end{eqnarray}
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