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