Question

Show that dy/dx = y(x-y) / x(x-y) if x = y log ( xy).

Answer 1

Given,

x=ylog(xy)

or,x=y[log(x)+log(y)]

or,x=y.log(x)+y.log(y)

Differentiating with respect to `x`

1=y/x+log(x).dy/dx+log(y).dy/dx+dy/dx

or,1-y/x=(dy/dx).[(y+x)/y]

or,(x-y)/x=(dy/dx).[(x+y)/y]

therefore,dy/dx=y(x-y)/x(x+y)