\[\boxed{\mathbf{848}\mathbf{.}}\]
\[1)\ f(x) = \sqrt{x^{2} + 2x - 1}\]
\[u = x^{2} + 2x - 1\ f(u) = \sqrt{u}:\]
\[f^{'}(x) = \left( x^{2} + 2x - 1 \right)^{'} \bullet \left( \sqrt{u} \right)^{'} =\]
\[= (2x + 2) \bullet \frac{1}{2\sqrt{u}} = \frac{x + 1}{\sqrt{x^{2} + 2x - 1}}.\]
\[2)\ f(x) = \sqrt[3]{\sin x}\]
\[u = \sin x\ \ f(u) = \sqrt[3]{u}:\]
\[f^{'}(x) = \left( \sin x \right)^{'} \bullet \left( u^{\frac{1}{3}} \right)^{'} =\]
\[= \cos x \bullet \frac{1}{3} \bullet u^{- \frac{2}{3}} = \frac{\cos x}{3\sqrt[3]{u^{2}}} =\]
\[= \frac{\cos x}{3\sqrt[3]{\sin^{2}x}}.\]
\[3)\ f(x) = \sqrt[4]{\cos x}\]
\[u = \cos x\ \text{\ f}(u) = \sqrt[4]{u}:\]
\[f^{'}(x) = \left( \cos x \right)^{'} \bullet \left( u^{\frac{1}{4}} \right)^{'} =\]
\[= - \sin x \bullet \frac{1}{4} \bullet u^{- \frac{3}{4}} = - \frac{\sin x}{4\sqrt[4]{u^{3}}} =\]
\[= - \frac{\sin x}{4\sqrt[4]{\cos^{3}x}}.\]
\[4)\ f(x) = \sqrt{\log_{2}x}\]
\[u = \log_{2}x\ \ f(u) = \sqrt{u}:\]
\[f^{'}(x) = \left( \log_{2}x \right)^{'} \bullet \left( \sqrt{u} \right)^{'} =\]
\[= \frac{1}{x \bullet \ln 2} \bullet \frac{1}{2\sqrt{u}} =\]
\[= \frac{1}{2x \bullet \ln 2 \bullet \sqrt{\log_{2}x}}.\]