Question

Найти производные функций, заданных в явном и неявном виде. а) y=(1+(x-1)/(x+1))^sqrt(2+x^2); б) x^3y-xy^3=sin(xy)

Answer 1

$a) y=( \frac{1+(x-1)}{(x+1)})^{ \sqrt{2+x^2} } \\y=(\frac{x}{x+1}) ^{ \sqrt{2+x^2} }\\ lny = ln[(\frac{x}{x+1})] ^{ \sqrt{2+x^2} }\\ \frac{y'}{y} =[\sqrt{2+x^2}ln(\frac{x}{x+1})]'=\\ = \frac{x}{ \sqrt{2+x^2} }ln( \frac{x}{x+1} )+\sqrt{2+x^2}(1+ \frac{1}{x} )( \frac{1}{(x+1)^2} ) \\ y'=[( \frac{1+(x-1)}{(x+1)})^{ \sqrt{2+x^2} }]*[\frac{x}{ \sqrt{2+x^2} }ln( \frac{x}{x+1} )+\sqrt{2+x^2}(1+ \frac{1}{x} )( \frac{1}{(x+1)^2} )]$

$x^3y-xy^3=sin(xy)\\ x^3y-xy^3-sin(xy)=0\\ F'_x=3x^2y-y^3-ycos(xy)\\ F'_y=x^3-3xy^2-xcos(xy)\\ -\frac{F'_x}{F'_y}= -\frac{3x^2y-y^3-ycos(xy)}{x^3-3xy^2-xcos(xy)}$