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$$$\cosh{\left(x \right)}$$$ 的二階導數
您的輸入
求$$$\frac{d^{2}}{dx^{2}} \left(\cosh{\left(x \right)}\right)$$$。
解答
求第一階導數 $$$\frac{d}{dx} \left(\cosh{\left(x \right)}\right)$$$
雙曲餘弦函數的導數為$$$\frac{d}{dx} \left(\cosh{\left(x \right)}\right) = \sinh{\left(x \right)}$$$：
$${\color{red}\left(\frac{d}{dx} \left(\cosh{\left(x \right)}\right)\right)} = {\color{red}\left(\sinh{\left(x \right)}\right)}$$因此，$$$\frac{d}{dx} \left(\cosh{\left(x \right)}\right) = \sinh{\left(x \right)}$$$。
接下來，$$$\frac{d^{2}}{dx^{2}} \left(\cosh{\left(x \right)}\right) = \frac{d}{dx} \left(\sinh{\left(x \right)}\right)$$$
雙曲正弦的導數為 $$$\frac{d}{dx} \left(\sinh{\left(x \right)}\right) = \cosh{\left(x \right)}$$$:
$${\color{red}\left(\frac{d}{dx} \left(\sinh{\left(x \right)}\right)\right)} = {\color{red}\left(\cosh{\left(x \right)}\right)}$$因此，$$$\frac{d}{dx} \left(\sinh{\left(x \right)}\right) = \cosh{\left(x \right)}$$$。
因此，$$$\frac{d^{2}}{dx^{2}} \left(\cosh{\left(x \right)}\right) = \cosh{\left(x \right)}$$$。
答案
$$$\frac{d^{2}}{dx^{2}} \left(\cosh{\left(x \right)}\right) = \cosh{\left(x \right)}$$$A
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