$$$\sin{\left(u \right)} - \cos{\left(u \right)}$$$ 的導數

和/差的導數等於導數的和/差：

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

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

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

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

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

化簡：

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

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