|
|
|
|
|
|
|
|
|
|
|
|
$$$\ln\left(x\right)$$$ 的二阶导数
您的输入
求$$$\frac{d^{2}}{dx^{2}} \left(\ln\left(x\right)\right)$$$。
解答
求一阶导数 $$$\frac{d}{dx} \left(\ln\left(x\right)\right)$$$
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，$$$\frac{d}{dx} \left(\ln\left(x\right)\right) = \frac{1}{x}$$$。
接下来，$$$\frac{d^{2}}{dx^{2}} \left(\ln\left(x\right)\right) = \frac{d}{dx} \left(\frac{1}{x}\right)$$$
应用幂次法则 $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$，其中 $$$n = -1$$$:
$${\color{red}\left(\frac{d}{dx} \left(\frac{1}{x}\right)\right)} = {\color{red}\left(- \frac{1}{x^{2}}\right)}$$因此，$$$\frac{d}{dx} \left(\frac{1}{x}\right) = - \frac{1}{x^{2}}$$$。
因此，$$$\frac{d^{2}}{dx^{2}} \left(\ln\left(x\right)\right) = - \frac{1}{x^{2}}$$$。
答案
$$$\frac{d^{2}}{dx^{2}} \left(\ln\left(x\right)\right) = - \frac{1}{x^{2}}$$$A
Please try a new game Rotatly