Expert: Abe Mantell - 1/23/2005

Question
I need to use logarithmic differentiation to find the derivative of (X^y)=(Y^x).  I run into trouble when I try to isolate y.  If I take "ln" of both sides I get (ylnx)=(xlny).  I can divide the right side by lnx, but I am stuck with the lny.
  Any help you could offer would be greatly appreciated.
     Thank You.

Answer
Hello Jayne,

Right, we cannot isolate y, so that is one reason for
logarithmic differentiation...

Differentiating yln(x)=xln(y) - via product & chain-rules - gives:
y(1/x)+y'ln(x)=x(1/y)y'+ln(y), now solve for y'
y'=(xyln(y)-y^2)/(ln(x)xy-x^2)

OK?

Abe