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$$$\ln\left(\frac{2}{x}\right)$$$ 的導數
您的輸入
求$$$\frac{d}{dx} \left(\ln\left(\frac{2}{x}\right)\right)$$$。
解答
函數 $$$\ln\left(\frac{2}{x}\right)$$$ 是兩個函數 $$$f{\left(u \right)} = \ln\left(u\right)$$$ 與 $$$g{\left(x \right)} = \frac{2}{x}$$$ 之複合 $$$f{\left(g{\left(x \right)} \right)}$$$。
應用鏈式法則 $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$：
$${\color{red}\left(\frac{d}{dx} \left(\ln\left(\frac{2}{x}\right)\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right) \frac{d}{dx} \left(\frac{2}{x}\right)\right)}$$自然對數的導數為 $$$\frac{d}{du} \left(\ln\left(u\right)\right) = \frac{1}{u}$$$：
$${\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right)\right)} \frac{d}{dx} \left(\frac{2}{x}\right) = {\color{red}\left(\frac{1}{u}\right)} \frac{d}{dx} \left(\frac{2}{x}\right)$$返回原變數：
$$\frac{\frac{d}{dx} \left(\frac{2}{x}\right)}{{\color{red}\left(u\right)}} = \frac{\frac{d}{dx} \left(\frac{2}{x}\right)}{{\color{red}\left(\frac{2}{x}\right)}}$$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 $$$c = 2$$$ 與 $$$f{\left(x \right)} = \frac{1}{x}$$$：
$$\frac{x {\color{red}\left(\frac{d}{dx} \left(\frac{2}{x}\right)\right)}}{2} = \frac{x {\color{red}\left(2 \frac{d}{dx} \left(\frac{1}{x}\right)\right)}}{2}$$套用冪次法則 $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$，取 $$$n = -1$$$：
$$x {\color{red}\left(\frac{d}{dx} \left(\frac{1}{x}\right)\right)} = x {\color{red}\left(- \frac{1}{x^{2}}\right)}$$因此，$$$\frac{d}{dx} \left(\ln\left(\frac{2}{x}\right)\right) = - \frac{1}{x}$$$。
答案
$$$\frac{d}{dx} \left(\ln\left(\frac{2}{x}\right)\right) = - \frac{1}{x}$$$A