Question

Найти производную x-y+arcsin(xy)=0

Answer 1

$x-y+arcsin(xy)=0 \ \ y'= -frac{F'_x}{F'_y} \ \ F_x'=1-0+ frac{1}{ sqrt{1-x^2} } \ \ F_y'=0-1+ frac{1}{ sqrt{1-y^2} }$

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