From Förberedande kurs i matematik 2

The expression is a sum of two terms which we differentiate one by one:

\displaystyle \begin{align} & {f}'\left( x \right)=\frac{d}{dx}\left( \cos x-\sin x \right) \\ & =\frac{d}{dx}\cos x-\frac{d}{dx}\sin x=-\sin x-\cos x \\ \end{align}