$$$e^{t} \cos{\left(t \right)}$$$ 的導數

將乘積法則 $$$\frac{d}{dt} \left(f{\left(t \right)} g{\left(t \right)}\right) = \frac{d}{dt} \left(f{\left(t \right)}\right) g{\left(t \right)} + f{\left(t \right)} \frac{d}{dt} \left(g{\left(t \right)}\right)$$$ 應用於 $$$f{\left(t \right)} = \cos{\left(t \right)}$$$ 和 $$$g{\left(t \right)} = e^{t}$$$：

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

餘弦函數的導數為 $$$\frac{d}{dt} \left(\cos{\left(t \right)}\right) = - \sin{\left(t \right)}$$$：

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

指數函數的導數為 $$$\frac{d}{dt} \left(e^{t}\right) = e^{t}$$$：

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

化簡：

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

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