Aus Online Mathematik Brückenkurs 2

When we see this expression, we should think "square root of something",

\displaystyle \sqrt{\bbox[#FFEEAA;,1.5pt]{\phantom{\cos x}}}\,\textrm{,}

and in order to differentiate it, we should first differentiate the outer function , "the square root of", with respect to its argument and, after that, multiply by the derivative of the inner functional expression \displaystyle \bbox[#FFEEAA;,1.5pt]{\phantom{\cos x}} = \cos x,

\displaystyle \frac{d}{dx}\,\sqrt{\bbox[#FFEEAA;,1.5pt]{\cos x}} = \frac{1}{2\sqrt{\bbox[#FFEEAA;,1.5pt]{\cos x}}}\cdot \bigl(\bbox[#FFEEAA;,1.5pt]{\cos x}\bigr)'\,,

where we have used the differentiation rule

\displaystyle \frac{d}{dx}\,\sqrt{x} = \frac{d}{dx}\,x^{1/2} = \tfrac{1}{2}x^{1/2-1} = \tfrac{1}{2}x^{-1/2} = \frac{1}{2\sqrt{x}}\,\textrm{.}

Thus, we obtain

\displaystyle \frac{d}{dx}\,\sqrt{\cos x} = \frac{1}{2\sqrt{\cos x}}\cdot (-\sin x) = -\frac{\sin x}{2\sqrt{\cos x}}\,\textrm{.}