$$$\ln^{4}\left(x\right)$$$ 的導數
您的輸入
求$$$\frac{d}{dx} \left(\ln^{4}\left(x\right)\right)$$$。
解答
函數 $$$\ln^{4}\left(x\right)$$$ 是兩個函數 $$$f{\left(u \right)} = u^{4}$$$ 與 $$$g{\left(x \right)} = \ln\left(x\right)$$$ 之複合 $$$f{\left(g{\left(x \right)} \right)}$$$。
應用鏈式法則 $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$：
$${\color{red}\left(\frac{d}{dx} \left(\ln^{4}\left(x\right)\right)\right)} = {\color{red}\left(\frac{d}{du} \left(u^{4}\right) \frac{d}{dx} \left(\ln\left(x\right)\right)\right)}$$套用冪次法則 $$$\frac{d}{du} \left(u^{n}\right) = n u^{n - 1}$$$，取 $$$n = 4$$$：
$${\color{red}\left(\frac{d}{du} \left(u^{4}\right)\right)} \frac{d}{dx} \left(\ln\left(x\right)\right) = {\color{red}\left(4 u^{3}\right)} \frac{d}{dx} \left(\ln\left(x\right)\right)$$返回原變數：
$$4 {\color{red}\left(u\right)}^{3} \frac{d}{dx} \left(\ln\left(x\right)\right) = 4 {\color{red}\left(\ln\left(x\right)\right)}^{3} \frac{d}{dx} \left(\ln\left(x\right)\right)$$自然對數的導數為 $$$\frac{d}{dx} \left(\ln\left(x\right)\right) = \frac{1}{x}$$$：
$$4 \ln^{3}\left(x\right) {\color{red}\left(\frac{d}{dx} \left(\ln\left(x\right)\right)\right)} = 4 \ln^{3}\left(x\right) {\color{red}\left(\frac{1}{x}\right)}$$因此，$$$\frac{d}{dx} \left(\ln^{4}\left(x\right)\right) = \frac{4 \ln^{3}\left(x\right)}{x}$$$。
答案
$$$\frac{d}{dx} \left(\ln^{4}\left(x\right)\right) = \frac{4 \ln^{3}\left(x\right)}{x}$$$A
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