z = x^(y-x) = exp((y-x)ln(x))
dz/dx = z*(((y - x)/x) - ln(x)) = 0
(y - x)/x) - ln(x) = 0
y = x x*ln(x) = x*(1 ln(x)) = x*ln(e*x)
Let Y = e*y and X = e*x
then : Y = X*ln(X)
The roots of this kind of equation cannot be expressed in terms of elementary function, but in term of the special function named the "Lambert's W function". See :
http://mathworld.wolfram.com/LambertW-Function.html
The result is : X = exp(W(Y))
Finally : x = exp(W(e*y)-1)
.
Remark : In case of some particular value of y, eventually this special function may be reduced to an elementary function.