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