From Förberedande kurs i matematik 2

Because the expression is a product of two factors, we use the product rule,

\displaystyle \begin{align} (\sin x\cdot\cos x)^{\prime } &= (\cos x)^{\prime }\cdot\sin x + \cos x\cdot (\sin x)^{\prime }\\[5pt] &= -\sin x\cdot\sin x + \cos x\cdot\cos x\\[5pt] &= -\sin^2\!x + \cos^2\!x\,\textrm{.} \end{align}

Using the formula for double angles, the answer can be simplified to \displaystyle \cos 2x\,.