$$$- \sin{\left(x \right)}$$$ 的二階導數
您的輸入
求$$$\frac{d^{2}}{dx^{2}} \left(- \sin{\left(x \right)}\right)$$$。
解答
求第一階導數 $$$\frac{d}{dx} \left(- \sin{\left(x \right)}\right)$$$
套用常數倍法則 $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$，使用 $$$c = -1$$$ 與 $$$f{\left(x \right)} = \sin{\left(x \right)}$$$：
$${\color{red}\left(\frac{d}{dx} \left(- \sin{\left(x \right)}\right)\right)} = {\color{red}\left(- \frac{d}{dx} \left(\sin{\left(x \right)}\right)\right)}$$正弦函數的導數為$$$\frac{d}{dx} \left(\sin{\left(x \right)}\right) = \cos{\left(x \right)}$$$：
$$- {\color{red}\left(\frac{d}{dx} \left(\sin{\left(x \right)}\right)\right)} = - {\color{red}\left(\cos{\left(x \right)}\right)}$$因此，$$$\frac{d}{dx} \left(- \sin{\left(x \right)}\right) = - \cos{\left(x \right)}$$$。
接下來，$$$\frac{d^{2}}{dx^{2}} \left(- \sin{\left(x \right)}\right) = \frac{d}{dx} \left(- \cos{\left(x \right)}\right)$$$
套用常數倍法則 $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$，使用 $$$c = -1$$$ 與 $$$f{\left(x \right)} = \cos{\left(x \right)}$$$：
$${\color{red}\left(\frac{d}{dx} \left(- \cos{\left(x \right)}\right)\right)} = {\color{red}\left(- \frac{d}{dx} \left(\cos{\left(x \right)}\right)\right)}$$餘弦函數的導數為 $$$\frac{d}{dx} \left(\cos{\left(x \right)}\right) = - \sin{\left(x \right)}$$$：
$$- {\color{red}\left(\frac{d}{dx} \left(\cos{\left(x \right)}\right)\right)} = - {\color{red}\left(- \sin{\left(x \right)}\right)}$$因此，$$$\frac{d}{dx} \left(- \cos{\left(x \right)}\right) = \sin{\left(x \right)}$$$。
因此，$$$\frac{d^{2}}{dx^{2}} \left(- \sin{\left(x \right)}\right) = \sin{\left(x \right)}$$$。
答案
$$$\frac{d^{2}}{dx^{2}} \left(- \sin{\left(x \right)}\right) = \sin{\left(x \right)}$$$A