a)
\(\begin{array}{l}
y = \left( {x + 2\sqrt x } \right)\left( {{x^2} + 4} \right) \Rightarrow y' = \left( {x + 2\sqrt x } \right)'\left( {{x^2} + 4} \right) + \left( {x + 2\sqrt x } \right)\left( {{x^2} + 4} \right)'\\
= \left( {1 + \frac{1}{{\sqrt x }}} \right)\left( {{x^2} + 4} \right) + 2x\left( {x + 2\sqrt x } \right) = 3{x^2} + 5x\sqrt x + \frac{4}{{\sqrt x }} + 4.
\end{array}\)
b)
\(\begin{array}{l}
y = {\cot ^2}\frac{2}{x} + \tan \frac{{x + 1}}{2} \Rightarrow y' = 2.\cot \frac{2}{x}\left( {\cot \frac{2}{x}} \right)' + \frac{{\left( {\frac{{x + 1}}{2}} \right)'}}{{{{\cos }^2}\frac{{x + 1}}{2}}}\\
= 2.\cot \frac{2}{x}\frac{{ - \left( {\frac{2}{x}} \right)}}{{{{\sin }^2}\frac{2}{x}}} + \frac{1}{{2c{\rm{o}}{{\rm{s}}^2}\frac{{x + 1}}{2}}} = 4\cot \frac{2}{x}.\frac{1}{{{x^2}{{\sin }^2}\frac{2}{x}}} + \frac{1}{{2c{\rm{o}}{{\rm{s}}^2}\frac{{x + 1}}{2}}}.
\end{array}\)
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