cos(ln(u))的导数

该计算器将求$$$\cos{\left(\ln\left(u\right) \right)}$$$的导数，并显示步骤。

您的输入

求$$$\frac{d}{du} \left(\cos{\left(\ln\left(u\right) \right)}\right)$$$。

解答

函数$$$\cos{\left(\ln\left(u\right) \right)}$$$是两个函数$$$f{\left(v \right)} = \cos{\left(v \right)}$$$和$$$g{\left(u \right)} = \ln\left(u\right)$$$的复合$$$f{\left(g{\left(u \right)} \right)}$$$。

应用链式法则 $$$\frac{d}{du} \left(f{\left(g{\left(u \right)} \right)}\right) = \frac{d}{dv} \left(f{\left(v \right)}\right) \frac{d}{du} \left(g{\left(u \right)}\right)$$$：

$${\color{red}\left(\frac{d}{du} \left(\cos{\left(\ln\left(u\right) \right)}\right)\right)} = {\color{red}\left(\frac{d}{dv} \left(\cos{\left(v \right)}\right) \frac{d}{du} \left(\ln\left(u\right)\right)\right)}$$

余弦函数的导数是$$$\frac{d}{dv} \left(\cos{\left(v \right)}\right) = - \sin{\left(v \right)}$$$：

$${\color{red}\left(\frac{d}{dv} \left(\cos{\left(v \right)}\right)\right)} \frac{d}{du} \left(\ln\left(u\right)\right) = {\color{red}\left(- \sin{\left(v \right)}\right)} \frac{d}{du} \left(\ln\left(u\right)\right)$$

返回到原变量：

$$- \sin{\left({\color{red}\left(v\right)} \right)} \frac{d}{du} \left(\ln\left(u\right)\right) = - \sin{\left({\color{red}\left(\ln\left(u\right)\right)} \right)} \frac{d}{du} \left(\ln\left(u\right)\right)$$

自然对数的导数为 $$$\frac{d}{du} \left(\ln\left(u\right)\right) = \frac{1}{u}$$$：

$$- \sin{\left(\ln\left(u\right) \right)} {\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right)\right)} = - \sin{\left(\ln\left(u\right) \right)} {\color{red}\left(\frac{1}{u}\right)}$$

因此，$$$\frac{d}{du} \left(\cos{\left(\ln\left(u\right) \right)}\right) = - \frac{\sin{\left(\ln\left(u\right) \right)}}{u}$$$。

答案

$$$\frac{d}{du} \left(\cos{\left(\ln\left(u\right) \right)}\right) = - \frac{\sin{\left(\ln\left(u\right) \right)}}{u}$$$A

Please try a new game Rotatly