$$$\sin{\left(a \right)} \cos{\left(x \right)} - \sin{\left(x \right)} \cos{\left(a \right)}$$$ 對 $$$x$$$ 的導數

和/差的導數等於導數的和/差：

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

套用常數倍法則 $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$，使用 $$$c = \cos{\left(a \right)}$$$ 與 $$$f{\left(x \right)} = \sin{\left(x \right)}$$$：

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

正弦函數的導數為$$$\frac{d}{dx} \left(\sin{\left(x \right)}\right) = \cos{\left(x \right)}$$$：

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

套用常數倍法則 $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$，使用 $$$c = \sin{\left(a \right)}$$$ 與 $$$f{\left(x \right)} = \cos{\left(x \right)}$$$：

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

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

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

化簡：

$$- \sin{\left(a \right)} \sin{\left(x \right)} - \cos{\left(a \right)} \cos{\left(x \right)} = - \cos{\left(a - x \right)}$$

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