$$$\frac{d^{4}}{dx^{4}} \left(\sin{\left(x \right)}\right)$$$ を求めよ

この計算機は$$$\frac{d^{4}}{dx^{4}} \left(\sin{\left(x \right)}\right)$$$を求め、手順を表示します。

入力内容

$$$\frac{d^{4}}{dx^{4}} \left(\sin{\left(x \right)}\right)$$$ を求めよ。

解答

一階導関数 $$$\frac{d}{dx} \left(\sin{\left(x \right)}\right)$$$ を求めよ

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、$$$\frac{d^{2}}{dx^{2}} \left(\sin{\left(x \right)}\right) = \frac{d}{dx} \left(\cos{\left(x \right)}\right)$$$

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、$$$\frac{d}{dx} \left(\cos{\left(x \right)}\right) = - \sin{\left(x \right)}$$$。

次に、$$$\frac{d^{3}}{dx^{3}} \left(\sin{\left(x \right)}\right) = \frac{d}{dx} \left(- \sin{\left(x \right)}\right)$$$

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 $$$f{\left(x \right)} = \sin{\left(x \right)}$$$ 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、$$$\frac{d^{4}}{dx^{4}} \left(\sin{\left(x \right)}\right) = \frac{d}{dx} \left(- \cos{\left(x \right)}\right)$$$

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、$$$\frac{d^{4}}{dx^{4}} \left(\sin{\left(x \right)}\right) = \sin{\left(x \right)}$$$。

解答

$$$\frac{d^{4}}{dx^{4}} \left(\sin{\left(x \right)}\right) = \sin{\left(x \right)}$$$A