$$$\cos{\left(n x \right)}$$$ 對 $$$x$$$ 的導數
您的輸入
求$$$\frac{d}{dx} \left(\cos{\left(n x \right)}\right)$$$。
解答
函數 $$$\cos{\left(n x \right)}$$$ 是兩個函數 $$$f{\left(u \right)} = \cos{\left(u \right)}$$$ 與 $$$g{\left(x \right)} = n x$$$ 之複合 $$$f{\left(g{\left(x \right)} \right)}$$$。
應用鏈式法則 $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$：
$${\color{red}\left(\frac{d}{dx} \left(\cos{\left(n x \right)}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\cos{\left(u \right)}\right) \frac{d}{dx} \left(n x\right)\right)}$$餘弦函數的導數為 $$$\frac{d}{du} \left(\cos{\left(u \right)}\right) = - \sin{\left(u \right)}$$$：
$${\color{red}\left(\frac{d}{du} \left(\cos{\left(u \right)}\right)\right)} \frac{d}{dx} \left(n x\right) = {\color{red}\left(- \sin{\left(u \right)}\right)} \frac{d}{dx} \left(n x\right)$$返回原變數：
$$- \sin{\left({\color{red}\left(u\right)} \right)} \frac{d}{dx} \left(n x\right) = - \sin{\left({\color{red}\left(n x\right)} \right)} \frac{d}{dx} \left(n x\right)$$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 $$$c = n$$$ 與 $$$f{\left(x \right)} = x$$$：
$$- \sin{\left(n x \right)} {\color{red}\left(\frac{d}{dx} \left(n x\right)\right)} = - \sin{\left(n x \right)} {\color{red}\left(n \frac{d}{dx} \left(x\right)\right)}$$套用冪次法則 $$$\frac{d}{dx} \left(x^{m}\right) = m x^{m - 1}$$$，取 $$$m = 1$$$，也就是 $$$\frac{d}{dx} \left(x\right) = 1$$$：
$$- n \sin{\left(n x \right)} {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} = - n \sin{\left(n x \right)} {\color{red}\left(1\right)}$$因此，$$$\frac{d}{dx} \left(\cos{\left(n x \right)}\right) = - n \sin{\left(n x \right)}$$$。
答案
$$$\frac{d}{dx} \left(\cos{\left(n x \right)}\right) = - n \sin{\left(n x \right)}$$$A