$$$- \sin{\left(x \right)}$$$ の二階導関数
入力内容
$$$\frac{d^{2}}{dx^{2}} \left(- \sin{\left(x \right)}\right)$$$ を求めよ。
解答
一階導関数 $$$\frac{d}{dx} \left(- \sin{\left(x \right)}\right)$$$ を求めよ
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 $$$f{\left(x \right)} = \sin{\left(x \right)}$$$ に対して適用します:
$${\color{red}\left(\frac{d}{dx} \left(- \sin{\left(x \right)}\right)\right)} = {\color{red}\left(- \frac{d}{dx} \left(\sin{\left(x \right)}\right)\right)}$$正弦関数の導関数は$$$\frac{d}{dx} \left(\sin{\left(x \right)}\right) = \cos{\left(x \right)}$$$:
$$- {\color{red}\left(\frac{d}{dx} \left(\sin{\left(x \right)}\right)\right)} = - {\color{red}\left(\cos{\left(x \right)}\right)}$$したがって、$$$\frac{d}{dx} \left(- \sin{\left(x \right)}\right) = - \cos{\left(x \right)}$$$。
次に、$$$\frac{d^{2}}{dx^{2}} \left(- \sin{\left(x \right)}\right) = \frac{d}{dx} \left(- \cos{\left(x \right)}\right)$$$
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 $$$f{\left(x \right)} = \cos{\left(x \right)}$$$ に対して適用します:
$${\color{red}\left(\frac{d}{dx} \left(- \cos{\left(x \right)}\right)\right)} = {\color{red}\left(- \frac{d}{dx} \left(\cos{\left(x \right)}\right)\right)}$$余弦関数の導関数は$$$\frac{d}{dx} \left(\cos{\left(x \right)}\right) = - \sin{\left(x \right)}$$$:
$$- {\color{red}\left(\frac{d}{dx} \left(\cos{\left(x \right)}\right)\right)} = - {\color{red}\left(- \sin{\left(x \right)}\right)}$$したがって、$$$\frac{d}{dx} \left(- \cos{\left(x \right)}\right) = \sin{\left(x \right)}$$$。
したがって、$$$\frac{d^{2}}{dx^{2}} \left(- \sin{\left(x \right)}\right) = \sin{\left(x \right)}$$$。
解答
$$$\frac{d^{2}}{dx^{2}} \left(- \sin{\left(x \right)}\right) = \sin{\left(x \right)}$$$A