|
|
|
|
|
|
|
|
|
|
|
|
$$$\ln\left(\frac{1}{x}\right)$$$的导数
您的输入
求$$$\frac{d}{dx} \left(\ln\left(\frac{1}{x}\right)\right)$$$。
解答
函数$$$\ln\left(\frac{1}{x}\right)$$$是两个函数$$$f{\left(u \right)} = \ln\left(u\right)$$$和$$$g{\left(x \right)} = \frac{1}{x}$$$的复合$$$f{\left(g{\left(x \right)} \right)}$$$。
应用链式法则 $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$：
$$\frac{d}{dx} \left(- \ln\left(x\right)\right) = \frac{d}{dx} \left(- \ln\left(x\right)\right)$$自然对数的导数为 $$$\frac{d}{du} \left(\ln\left(u\right)\right) = \frac{1}{u}$$$：
$$\frac{d}{dx} \left(- \ln\left(x\right)\right) = \frac{d}{dx} \left(- \ln\left(x\right)\right)$$返回到原变量：
$$\frac{d}{dx} \left(- \ln\left(x\right)\right) = \frac{d}{dx} \left(- \ln\left(x\right)\right)$$对 $$$c = -1$$$ 和 $$$f{\left(x \right)} = \ln\left(x\right)$$$ 应用常数倍法则 $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$：
$${\color{red}\left(\frac{d}{dx} \left(- \ln\left(x\right)\right)\right)} = {\color{red}\left(- \frac{d}{dx} \left(\ln\left(x\right)\right)\right)}$$自然对数的导数为 $$$\frac{d}{dx} \left(\ln\left(x\right)\right) = \frac{1}{x}$$$：
$$- {\color{red}\left(\frac{d}{dx} \left(\ln\left(x\right)\right)\right)} = - {\color{red}\left(\frac{1}{x}\right)}$$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