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