$$$\sin{\left(x \right)} - \cos{\left(x \right)}$$$的导数

和/差的导数等于导数的和/差：

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

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

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

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

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

化简：

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

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