$$$\csc{\left(x \right)}$$$ の二階導関数
入力内容
$$$\frac{d^{2}}{dx^{2}} \left(\csc{\left(x \right)}\right)$$$ を求めよ。
解答
一階導関数 $$$\frac{d}{dx} \left(\csc{\left(x \right)}\right)$$$ を求めよ
コセカント関数の導関数は$$$\frac{d}{dx} \left(\csc{\left(x \right)}\right) = - \cot{\left(x \right)} \csc{\left(x \right)}$$$です:
$${\color{red}\left(\frac{d}{dx} \left(\csc{\left(x \right)}\right)\right)} = {\color{red}\left(- \cot{\left(x \right)} \csc{\left(x \right)}\right)}$$したがって、$$$\frac{d}{dx} \left(\csc{\left(x \right)}\right) = - \cot{\left(x \right)} \csc{\left(x \right)}$$$。
次に、$$$\frac{d^{2}}{dx^{2}} \left(\csc{\left(x \right)}\right) = \frac{d}{dx} \left(- \cot{\left(x \right)} \csc{\left(x \right)}\right)$$$
定数倍の法則 $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ を $$$c = -1$$$ と $$$f{\left(x \right)} = \cot{\left(x \right)} \csc{\left(x \right)}$$$ に対して適用します:
$${\color{red}\left(\frac{d}{dx} \left(- \cot{\left(x \right)} \csc{\left(x \right)}\right)\right)} = {\color{red}\left(- \frac{d}{dx} \left(\cot{\left(x \right)} \csc{\left(x \right)}\right)\right)}$$積の微分法 $$$\frac{d}{dx} \left(f{\left(x \right)} g{\left(x \right)}\right) = \frac{d}{dx} \left(f{\left(x \right)}\right) g{\left(x \right)} + f{\left(x \right)} \frac{d}{dx} \left(g{\left(x \right)}\right)$$$ を $$$f{\left(x \right)} = \cot{\left(x \right)}$$$ と $$$g{\left(x \right)} = \csc{\left(x \right)}$$$ に適用する:
$$- {\color{red}\left(\frac{d}{dx} \left(\cot{\left(x \right)} \csc{\left(x \right)}\right)\right)} = - {\color{red}\left(\frac{d}{dx} \left(\cot{\left(x \right)}\right) \csc{\left(x \right)} + \cot{\left(x \right)} \frac{d}{dx} \left(\csc{\left(x \right)}\right)\right)}$$コセカント関数の導関数は$$$\frac{d}{dx} \left(\csc{\left(x \right)}\right) = - \cot{\left(x \right)} \csc{\left(x \right)}$$$です:
$$- \cot{\left(x \right)} {\color{red}\left(\frac{d}{dx} \left(\csc{\left(x \right)}\right)\right)} - \csc{\left(x \right)} \frac{d}{dx} \left(\cot{\left(x \right)}\right) = - \cot{\left(x \right)} {\color{red}\left(- \cot{\left(x \right)} \csc{\left(x \right)}\right)} - \csc{\left(x \right)} \frac{d}{dx} \left(\cot{\left(x \right)}\right)$$余接関数の導関数は$$$\frac{d}{dx} \left(\cot{\left(x \right)}\right) = - \csc^{2}{\left(x \right)}$$$:
$$\cot^{2}{\left(x \right)} \csc{\left(x \right)} - \csc{\left(x \right)} {\color{red}\left(\frac{d}{dx} \left(\cot{\left(x \right)}\right)\right)} = \cot^{2}{\left(x \right)} \csc{\left(x \right)} - \csc{\left(x \right)} {\color{red}\left(- \csc^{2}{\left(x \right)}\right)}$$簡単化せよ:
$$\cot^{2}{\left(x \right)} \csc{\left(x \right)} + \csc^{3}{\left(x \right)} = \left(-1 + \frac{2}{\sin^{2}{\left(x \right)}}\right) \csc{\left(x \right)}$$したがって、$$$\frac{d}{dx} \left(- \cot{\left(x \right)} \csc{\left(x \right)}\right) = \left(-1 + \frac{2}{\sin^{2}{\left(x \right)}}\right) \csc{\left(x \right)}$$$。
したがって、$$$\frac{d^{2}}{dx^{2}} \left(\csc{\left(x \right)}\right) = \left(-1 + \frac{2}{\sin^{2}{\left(x \right)}}\right) \csc{\left(x \right)}$$$。
解答
$$$\frac{d^{2}}{dx^{2}} \left(\csc{\left(x \right)}\right) = \left(-1 + \frac{2}{\sin^{2}{\left(x \right)}}\right) \csc{\left(x \right)}$$$A