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$$$\ln\left(x\right)$$$ 的二階導數
您的輸入
求$$$\frac{d^{2}}{dx^{2}} \left(\ln\left(x\right)\right)$$$。
解答
求第一階導數 $$$\frac{d}{dx} \left(\ln\left(x\right)\right)$$$
自然對數的導數為 $$$\frac{d}{dx} \left(\ln\left(x\right)\right) = \frac{1}{x}$$$：
$${\color{red}\left(\frac{d}{dx} \left(\ln\left(x\right)\right)\right)} = {\color{red}\left(\frac{1}{x}\right)}$$因此，$$$\frac{d}{dx} \left(\ln\left(x\right)\right) = \frac{1}{x}$$$。
接下來，$$$\frac{d^{2}}{dx^{2}} \left(\ln\left(x\right)\right) = \frac{d}{dx} \left(\frac{1}{x}\right)$$$
套用冪次法則 $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$，取 $$$n = -1$$$：
$${\color{red}\left(\frac{d}{dx} \left(\frac{1}{x}\right)\right)} = {\color{red}\left(- \frac{1}{x^{2}}\right)}$$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
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