Question

Помогите 2-4 задания пжл

Answer 1

$2); ; y=frac{2}{3}sqrt{(arctge^{x})^3}=frac{2}{3}(arctge^{x})^{frac{3}{2}}\\y'=frac{2}{3}cdot frac{3}{2}cdot (arctge^{x})^{frac{1}{2}}cdot frac{e^{x}}{1+e^{2x}}=frac{e^{x}}{1+e^{2x}}cdot sqrt{arctge^{x}}\\3); ; y=ln, arcsin sqrt{1-e^{2x}}\\y'=frac{1}{arcsinsqrt{1-e^{2x}}}cdot frac{1}{sqrt{1-(1-e^{2x})}}cdot frac{-2e^{2x}}{2sqrt{1-e^{2x}}}=frac{-e^{2x}}{sqrt{e^{2x}cdot (1-e^{2x})}cdot arcsinsqrt{1-e^{2x}}}$

$4); ; y=sqrt{frac{2}{3}}cdot arctgfrac{3x-1}{sqrt{6x}}\\y'=sqrt{frac{2}{3}}cdot frac{1}{1+frac{(3x-1)^2}{6x}}cdot frac{3cdot sqrt{6x}-(3x-1)cdot sqrt6cdot frac{1}{2sqrt{x}}}{6x}=\\=sqrt{frac{2}{3}}cdot frac{6x}{9x^2+1}cdot frac{sqrt6(3x+1)}{12xsqrt{x}}=frac{3x+1}{sqrt{x}(9x^2+1)}$