Question

Найти производную функции.
y=sqrt(x/(x+1))

Answer 1

$y= sqrt{ frac{x}{x+1}}=( frac{x}{x+1}})^{0.5}$
$y'=0.5*( frac{x}{x+1}})^{-0.5}*frac{x'*(x+1)-x(x+1)'}{(x+1)^{2}}=0.5*( frac{x}{x+1})^{-0.5}*frac{x+1-x}{(x+1)^{2}}=frac{1}{2 sqrt{frac{x}{x+1}}*(x+1)^{2}}=frac{ sqrt{x+1}}{2 sqrt{x}*(x+1)^{2}}=frac{1}{2 sqrt{x(x+1)}*(x+1)}=frac{1}{2 sqrt{x^{2}+x}*(x+1)}$