Question

Найдите производную функции у=(Х-1 ) ⁄ Х

Answer 1

$y(x)=\frac{x-1}{x}=\frac{x}{x}-\frac{1}{x}=1-x^{-1}\\\\ y'(x)=[1-x^{-1}]'=[1]'-[x^{-1}]'=0-(-1)*x^{-1-1}=x^{-2}=\frac{1}{x^2}\\\\ (\frac{u}{v})=\frac{u'*v-u*v'}{v^2}\\\\ y'(x)=[\frac{x-1}{x}]'=\frac{[x-1]'*x-(x-1)*[x]'}{x^2}=\\\\ =\frac{[(x)'-(1)']*x-(x-1)*[1]}{x^2}=\frac{[1-0]*x-(x-1)}{x^2}=\\\\ =\frac{x-x+1}{x^2}=\frac{1}{x^2}$