How do you use implicit differentiation to find #(dy)/(dx)# given #2xy-y^2=3#?

1 Answer
Oct 9, 2016

#(dy)/(dx)= y/(y-1)#

Explanation:

Given
#color(white)("XXX")color(red)(2xy)-color(blue)(y^2) = color(green)3#

Note 1:
#color(white)("XXX")color(red)((d(2xy))/(dx)) = 2x * (dy)/(dx) + y * (d(2x))/(dx) =color(red)( 2(dy)/(dx)+2y)#

Note 2:
#color(white)("XXX")color(blue)((d(y^2))/(dx)) = color(blue)(2y(dy)/(dx))#

Note 3:
#color(white)("XXX")color(green)((d(3))/(dx))=color(green)(0)#

Therefore #2xy-y^2=3#

#rArr color(red)(2(dy)/(dx)+2y)-color(blue)(2y(dy)/(dx))=color(green)(0)#

#rArr (dy)/(dx)-y(dy)/(dx)= -y#

#rArr (dy)/(dx)(1-y) = -y#

#rArr (dy)/(dx)= y/(y-1)#