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$$$\ln\left(\frac{a^{2}}{x^{2}}\right)$$$ 关于 $$$x$$$ 的导数
相关计算器: 对数求导法计算器, 带步骤的隐函数求导计算器
您的输入
求$$$\frac{d}{dx} \left(\ln\left(\frac{a^{2}}{x^{2}}\right)\right)$$$。
解答
函数$$$\ln\left(\frac{a^{2}}{x^{2}}\right)$$$是两个函数$$$f{\left(u \right)} = \ln\left(u\right)$$$和$$$g{\left(x \right)} = \frac{a^{2}}{x^{2}}$$$的复合$$$f{\left(g{\left(x \right)} \right)}$$$。
应用链式法则 $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$：
$${\color{red}\left(\frac{d}{dx} \left(\ln\left(\frac{a^{2}}{x^{2}}\right)\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right) \frac{d}{dx} \left(\frac{a^{2}}{x^{2}}\right)\right)}$$自然对数的导数为 $$$\frac{d}{du} \left(\ln\left(u\right)\right) = \frac{1}{u}$$$：
$${\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right)\right)} \frac{d}{dx} \left(\frac{a^{2}}{x^{2}}\right) = {\color{red}\left(\frac{1}{u}\right)} \frac{d}{dx} \left(\frac{a^{2}}{x^{2}}\right)$$返回到原变量：
$$\frac{\frac{d}{dx} \left(\frac{a^{2}}{x^{2}}\right)}{{\color{red}\left(u\right)}} = \frac{\frac{d}{dx} \left(\frac{a^{2}}{x^{2}}\right)}{{\color{red}\left(\frac{a^{2}}{x^{2}}\right)}}$$对 $$$c = a^{2}$$$ 和 $$$f{\left(x \right)} = \frac{1}{x^{2}}$$$ 应用常数倍法则 $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$：
$$\frac{x^{2} {\color{red}\left(\frac{d}{dx} \left(\frac{a^{2}}{x^{2}}\right)\right)}}{a^{2}} = \frac{x^{2} {\color{red}\left(a^{2} \frac{d}{dx} \left(\frac{1}{x^{2}}\right)\right)}}{a^{2}}$$应用幂次法则 $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$，其中 $$$n = -2$$$:
$$x^{2} {\color{red}\left(\frac{d}{dx} \left(\frac{1}{x^{2}}\right)\right)} = x^{2} {\color{red}\left(- \frac{2}{x^{3}}\right)}$$因此，$$$\frac{d}{dx} \left(\ln\left(\frac{a^{2}}{x^{2}}\right)\right) = - \frac{2}{x}$$$。
答案
$$$\frac{d}{dx} \left(\ln\left(\frac{a^{2}}{x^{2}}\right)\right) = - \frac{2}{x}$$$A