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You don't consider the right equation to solve:

y=function('y')(x)
dy = diff(y)
derivative = solve(diff(y==exp(x*y)), dy)
derivative[0].subs(exp(x*y)==y)

This yields

diff(y(x), x) == -y(x)^2/(x*y(x) - 1)

that is, $$y'=\frac{y^2}{1-xy},$$ in agreement with your hand calculation.