$$$\cos{\left(\frac{2 \ln\left(x\right)}{3} \right)}$$$的导数

函数$$$\cos{\left(\frac{2 \ln\left(x\right)}{3} \right)}$$$是两个函数$$$f{\left(u \right)} = \cos{\left(u \right)}$$$和$$$g{\left(x \right)} = \frac{2 \ln\left(x\right)}{3}$$$的复合$$$f{\left(g{\left(x \right)} \right)}$$$。

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

$${\color{red}\left(\frac{d}{dx} \left(\cos{\left(\frac{2 \ln\left(x\right)}{3} \right)}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\cos{\left(u \right)}\right) \frac{d}{dx} \left(\frac{2 \ln\left(x\right)}{3}\right)\right)}$$

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

$${\color{red}\left(\frac{d}{du} \left(\cos{\left(u \right)}\right)\right)} \frac{d}{dx} \left(\frac{2 \ln\left(x\right)}{3}\right) = {\color{red}\left(- \sin{\left(u \right)}\right)} \frac{d}{dx} \left(\frac{2 \ln\left(x\right)}{3}\right)$$

返回到原变量：

$$- \sin{\left({\color{red}\left(u\right)} \right)} \frac{d}{dx} \left(\frac{2 \ln\left(x\right)}{3}\right) = - \sin{\left({\color{red}\left(\frac{2 \ln\left(x\right)}{3}\right)} \right)} \frac{d}{dx} \left(\frac{2 \ln\left(x\right)}{3}\right)$$

对 $$$c = \frac{2}{3}$$$ 和 $$$f{\left(x \right)} = \ln\left(x\right)$$$ 应用常数倍法则 $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$：

$$- \sin{\left(\frac{2 \ln\left(x\right)}{3} \right)} {\color{red}\left(\frac{d}{dx} \left(\frac{2 \ln\left(x\right)}{3}\right)\right)} = - \sin{\left(\frac{2 \ln\left(x\right)}{3} \right)} {\color{red}\left(\frac{2 \frac{d}{dx} \left(\ln\left(x\right)\right)}{3}\right)}$$

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

$$- \frac{2 \sin{\left(\frac{2 \ln\left(x\right)}{3} \right)} {\color{red}\left(\frac{d}{dx} \left(\ln\left(x\right)\right)\right)}}{3} = - \frac{2 \sin{\left(\frac{2 \ln\left(x\right)}{3} \right)} {\color{red}\left(\frac{1}{x}\right)}}{3}$$

因此，$$$\frac{d}{dx} \left(\cos{\left(\frac{2 \ln\left(x\right)}{3} \right)}\right) = - \frac{2 \sin{\left(\frac{2 \ln\left(x\right)}{3} \right)}}{3 x}$$$。