$$$e^{x} \cos{\left(x \right)}$$$的导数

对 $$$f{\left(x \right)} = \cos{\left(x \right)}$$$ 和 $$$g{\left(x \right)} = e^{x}$$$ 应用乘积法则 $$$\frac{d}{dx} \left(f{\left(x \right)} g{\left(x \right)}\right) = \frac{d}{dx} \left(f{\left(x \right)}\right) g{\left(x \right)} + f{\left(x \right)} \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:

$${\color{red}\left(\frac{d}{dx} \left(e^{x} \cos{\left(x \right)}\right)\right)} = {\color{red}\left(\frac{d}{dx} \left(\cos{\left(x \right)}\right) e^{x} + \cos{\left(x \right)} \frac{d}{dx} \left(e^{x}\right)\right)}$$

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

$$e^{x} {\color{red}\left(\frac{d}{dx} \left(\cos{\left(x \right)}\right)\right)} + \cos{\left(x \right)} \frac{d}{dx} \left(e^{x}\right) = e^{x} {\color{red}\left(- \sin{\left(x \right)}\right)} + \cos{\left(x \right)} \frac{d}{dx} \left(e^{x}\right)$$

指数函数的导数为 $$$\frac{d}{dx} \left(e^{x}\right) = e^{x}$$$：

$$- e^{x} \sin{\left(x \right)} + \cos{\left(x \right)} {\color{red}\left(\frac{d}{dx} \left(e^{x}\right)\right)} = - e^{x} \sin{\left(x \right)} + \cos{\left(x \right)} {\color{red}\left(e^{x}\right)}$$

化简：

$$- e^{x} \sin{\left(x \right)} + e^{x} \cos{\left(x \right)} = \sqrt{2} e^{x} \cos{\left(x + \frac{\pi}{4} \right)}$$

因此，$$$\frac{d}{dx} \left(e^{x} \cos{\left(x \right)}\right) = \sqrt{2} e^{x} \cos{\left(x + \frac{\pi}{4} \right)}$$$。