$$$e^{- x} \sin{\left(x \right)}$$$ 在 $$$x = c$$$ 处的导数

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

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

函数$$$e^{- x}$$$是两个函数$$$f{\left(u \right)} = e^{u}$$$和$$$g{\left(x \right)} = - x$$$的复合$$$f{\left(g{\left(x \right)} \right)}$$$。

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

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

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

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

返回到原变量：

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

对 $$$c = -1$$$ 和 $$$f{\left(x \right)} = x$$$ 应用常数倍法则 $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$：

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

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

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

应用幂法则 $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$，取 $$$n = 1$$$，也就是说，$$$\frac{d}{dx} \left(x\right) = 1$$$：

$$- e^{- x} \sin{\left(x \right)} {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} + e^{- x} \cos{\left(x \right)} = - e^{- x} \sin{\left(x \right)} {\color{red}\left(1\right)} + e^{- x} \cos{\left(x \right)}$$

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

最后，在$$$x = c$$$处计算导数的值。

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