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$$$\cosh{\left(x \right)}$$$ の二階導関数
入力内容
$$$\frac{d^{2}}{dx^{2}} \left(\cosh{\left(x \right)}\right)$$$ を求めよ。
解答
一階導関数 $$$\frac{d}{dx} \left(\cosh{\left(x \right)}\right)$$$ を求めよ
双曲線余弦の導関数は$$$\frac{d}{dx} \left(\cosh{\left(x \right)}\right) = \sinh{\left(x \right)}$$$です:
$${\color{red}\left(\frac{d}{dx} \left(\cosh{\left(x \right)}\right)\right)} = {\color{red}\left(\sinh{\left(x \right)}\right)}$$したがって、$$$\frac{d}{dx} \left(\cosh{\left(x \right)}\right) = \sinh{\left(x \right)}$$$。
次に、$$$\frac{d^{2}}{dx^{2}} \left(\cosh{\left(x \right)}\right) = \frac{d}{dx} \left(\sinh{\left(x \right)}\right)$$$
双曲線正弦関数の導関数は$$$\frac{d}{dx} \left(\sinh{\left(x \right)}\right) = \cosh{\left(x \right)}$$$:
$${\color{red}\left(\frac{d}{dx} \left(\sinh{\left(x \right)}\right)\right)} = {\color{red}\left(\cosh{\left(x \right)}\right)}$$したがって、$$$\frac{d}{dx} \left(\sinh{\left(x \right)}\right) = \cosh{\left(x \right)}$$$。
したがって、$$$\frac{d^{2}}{dx^{2}} \left(\cosh{\left(x \right)}\right) = \cosh{\left(x \right)}$$$。
解答
$$$\frac{d^{2}}{dx^{2}} \left(\cosh{\left(x \right)}\right) = \cosh{\left(x \right)}$$$A
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