Question

( x^{y}+y^{x}=1 )
Let ( x^{y}=t ; y log x=log t )
( Rightarrow frac{y}{x}+log x frac{d y}{d x}=frac{1}{t} frac{d t}{d x} )
( Rightarrow frac{d t}{d x}=x^{y}left(frac{y}{x}+log x frac{d y}{d x}right) )
Let ( y^{x}=Delta j-x log y=log s )
( Rightarrow ) leas..
( log y+frac{x}{y} frac{d y}{d x}=frac{1}{5} frac{d s}{d x} )
( Rightarrow frac{d s}{d x}=y^{x}left(log y+frac{x}{y} frac{d y}{d x}right) )
from ( (1), quad & x^{y}+y^{x}=1 Rightarrow t+s=1 )
( Rightarrow frac{d t}{d x}+frac{d s}{d x}=0 )
( Rightarrow y x^{y-1}+x^{y} log x d y+y^{x} log y+y^{x-1} x frac{d y}{d x}=0 )
( Rightarrow frac{d y}{d x}=frac{-left(y^{x} log y+y x^{y-1}right)}{x^{y} log x+x y^{x-1}} )

# . If xy + y = 1, find ay. dx

Solution