If xy log(x+y)=1, prove that dy/dx= -y(x2y+x+y)/x(xy2+x+y) - Maths - Continuity and Differentiability

Question

If xy log(x+y)=1, prove that dy/dx=
-y(x2y+x+y)/x(xy2+x+y)

Priyanka Kedia · Accepted Answer

Given, xylogx+y=1 1Differentiate with respect to x,
xddxylogx+y+ylogx+yddxx=ddx1⇒xyddxlogx+y+logx+yddxy+ylogx+y1=0⇒xy1x+y1+dydx+logx+ydydx+ylogx+y=0⇒xyx+y+xyx+ydydx+xlogx+ydydx+ylogx+y=0⇒xyx+y+xyx+ydydx+xxydydx+yxy=0
from equation 1, logx+y=1xy⇒xyx+y+xyx+ydydx+1ydydx+1x=0
⇒xyx+y+1x+xyx+y+1ydydx=0⇒x2y+x+yxx+y+xy2+x+yyx+ydydx=0⇒dydx=-x2y+x+yxx+yxy2+x+yyx+y⇒dydx=-yx2y+x+yxxy2+x+yHence
proved