Решить 2.1 ,2.2 ......

Только производные:
2.1
$y=sin frac{x}{2}+cos frac{x}{2} y= frac{1}{2} cos frac{x}{2}- frac{1}{2}sin frac{x}{2}$

2.2
$y= sqrt[3]{(4+3x)^2}=(4+3x)^{ frac{2}{3} } y= frac{2}{3}(4+3x)^{ frac{2}{3}- frac{3}{3} }*3=2(4+3x)^{- frac{1}{3} }= frac{2}{ sqrt[3]{4+3x} }$

2.3
$y= sqrt{2x-sin2x} y= frac{1}{2 sqrt{2x-sin2x} }*(2-2cos2x)= frac{2(1-cos2x)}{2 sqrt{2x-sin2x} }= frac{1-cos2x}{ sqrt{2x-sin2x} }$

2.4
$y= sqrt[4]{1+cos^{2}x} =(1+cos^{2}x)^{ frac{1}{4} } y= frac{1}{4}(1+cos^2x)^{ frac{1}{4}- frac{4}{4} }*(2cosx*(-sinx))= = frac{1}{4}(1+cos^2x)^{- frac{3}{4} }*(-2sinxcosx)=- frac{sin2x}{4 sqrt[4]{(1+cos^2x)^3} }$

2.5
$y=sin sqrt{x} y=cos sqrt{x} * frac{1}{2 sqrt{x} } = frac{cos sqrt{x} }{2 sqrt{x} }$

2.6
$y= frac{1}{(1+cos4x)^5}=(1+cos4x)^{- 5 } y=-5(1+cos4x)^{-5-1}*(-4sin4x)= frac{20sin4x}{(1+cos4x)^6}$

2.7
не понятно условие

2.8
$y=x sqrt{x^2-1} y=x*( sqrt{x^2-1} )+x*( sqrt{x^2-1} )= sqrt{x^2-1}+ frac{x}{2 sqrt{x^2-1} }*2x= = sqrt{x^2-1}+ frac{x^2}{ sqrt{x^2-1} }= frac{x^2-1+x^2}{ sqrt{x^2-1} }= frac{2x^2-1}{ sqrt{x^2-1} }$

2.9
$y= sqrt{4x+sin4x} y= frac{1}{2 sqrt{4x+sin4x} }*(4+4cos4x)= frac{4(1+cos4x)}{2 sqrt{4x+sin4x} }= = frac{2(1+cos4x)}{ sqrt{4x+sin4x} }= frac{2+2cos4x}{ sqrt{4x+sin4x} }$

2.10
$y= frac{1+sin2x}{1-sin2x} y= frac{(1+sin2x)*(1-sin2x)-(1+sin2x)*(1-sin2x)}{(1-sin2x)^2}= =frac{2cos2x(1-sin2x)-(1+sin2x)*(-2cos2x)}{(1-sin2x)^2}= = frac{2cos2x-2sin2xcos2x+2cos2x+2sin2xcos2x}{(1-sin2x)^2}= = frac{4cos2x}{(1-sin2x)^2}$

2.11
$y= sqrt{ frac{x}{2}-sin frac{x}{2} } y= frac{1}{2 sqrt{ frac{x}{2}-sin frac{x}{2} } }*( frac{1}{2}- frac{1}{2}cos frac{x}{2} )= = frac{ frac{1}{2}(1-cos frac{x}{2} ) }{2 sqrt{ frac{x}{2}-sin frac{x}{2} } } = frac{1-cos frac{x}{2} }{4 sqrt{ frac{x}{2}-sin frac{x}{2} } }$

2.12
$y= sqrt{1+cos^{2}x^2} y= frac{1}{2 sqrt{1+cos^2x^2} }*(1+2cosx^2*(-sinx^2)*2x)= = frac{1-2xsin2x^2}{2 sqrt{1+cos^2x^2} }$

2.13
$y= frac{ sqrt{4x+1} }{x^2} y= frac{( sqrt{4x+1} )*x^2- sqrt{4x+1}*(x^2) }{(x^2)^2}= frac{ (frac{1}{2 sqrt{4x+1} }*4)*x^2-2x sqrt{4x+1} }{x^4}= = frac{ frac{2x^2}{ sqrt{4x+1} }-2x sqrt{4x+1} }{x^4}= frac{ frac{2x^2-2x(4x+1)}{ sqrt{4x+1} } }{x^4}= frac{2x^2-8x^2-2x}{x^4 sqrt{4x+1} }= = frac{-6x^2-2x}{x^4 sqrt{4x+1} }= frac{-x(6x+2)}{x^4 sqrt{4x+1} }=- frac{6x+2}{x^3 sqrt{4x+1} }$

2.14
$y=sin^2x^3 y=2sinx^3*cosx^3*3x^2=3x^2sin2x^3$

2.15
$y=tgx+ frac{2}{3}tg^3x + frac{1}{5}tg^5x y= frac{1}{cos^2x} + frac{2}{3}*3tg^2x* frac{1}{cos^2x}+ frac{1}{5}*5tg^4x* frac{1}{cos^2x}= = frac{1}{cos^2x}+ frac{2tg^2x}{cos^2x}+ frac{5tg^4x}{cos^2x}= frac{1+2tg^2x+5tg^4x}{cos^2x}$

2.16
$y=sinx^3+cos^3x y=cosx^3*3x^2+3cos^2x*(-sinx)=3x^2cosx^3-3cos^2xsinx$

2.17
$y=(1+ frac{1}{ sqrt[3]{x} } )^3=(1+x^{- frac{1}{3} })^3 y=3(1+ frac{1}{ sqrt[3]{x} } )^2*(- frac{1}{3}x^{- frac{4}{3} } )=- frac{(1+ frac{1}{ sqrt[3]{x} })^2 }{ sqrt[3]{x^4} } =- frac{(1+ frac{1}{ sqrt[3]{x} } )^2}{x sqrt[3]{x} }$

2.18
$y= frac{cosx}{1+2sinx} y= frac{(cosx)(1+2sinx)-cosx*(1+2sinx)}{(1+2sinx)^2}= frac{-sinx(1+2sinx)-cosx*(2cosx)}{(1+2sinx)^2}= = frac{-sinx-2sin^2x-2cos^2x}{(1+2sinx)^2}= frac{-sinx-2(sin^2x+cos^2x)}{(1+2sinx)^2}= = frac{-sinx-2}{(1+2sinx)^2}=- frac{sinx+2}{(1+2sinx)^2}$

2.19
$y=x- frac{2}{x}- frac{1}{3x^3}=x-2x^{-1}- frac{1}{3} x^{-3} y=1-2*(- frac{1}{x^2} )- frac{1}{3}*(-3x^{-4})= =1+ frac{2}{x^2}+ frac{1}{x^4}= (1+ frac{1}{x^2} )^2$

2.20
$y= frac{x^2-1}{x^2+1} y= frac{(x^2-1)*(x^2+1)-(x^2-1)*(x^2+1)}{(x^2+1)^2}= frac{2x(x^2+1)-2x(x^2-1)}{(x^2+1)^2}= = frac{2x(x^2+1-x^2+1)}{(x^2+1)^2} = frac{4x}{(x^2+1)^2}$