Question

Найдите производную третьего порядка. f(x)=1/x+sinx

Answer 1

$f(x)=frac{1}{x+sinx}; ; ,quad (frac{1}u})'=-frac{u'}{u^2}\\f'(x)=-frac{(x+sinx)'}{(x+sinx)^2}=-frac{1+cosx}{(x+sinx)^2}\\f''(x)=-frac{-sinx(x+sinx)^2-(1+cosx)cdot 2(x+sinx)cdot (1+cosx)}{(x+sinx)^4}=frac{sinxcdot (x+sinx)+2cdot (1+cosx)^2}{(x+sinx)^3}\\f'''(x)=frac{1}{(x+sinx)^3}cdot Big [Big (cosx(x+sinx)+sinx(1+cosx)-\\-4(1+cosx)cdot sinxBig )(x+sinx)^3-Big (sinxcdot (x+sinx)+2cdot (1+cosx)^2Big )times \\times 3(x+sinx)^2cdot (1+cosx)Big ]$