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Unfortunately, this is pretty hard to do in Sage. Even this attempt (eventually) fails:

sage: y = function('y',x)
sage: f = y*x==1; f
x*y(x) == 1
sage: f.derivative(x)
x*D[0](y)(x) + y(x) == 0
sage: g = f.derivative(x)
sage: g.operands()[0].operands()[0].operands()[1]
sage: h = g.solve(g.operands()[0].operands()[0].operands()[1])[0]; h
D[0](y)(x) == -y(x)/x
sage: implicit_plot(h.rhs(),(x,-1,1),(y,-1,1))

I don't know that this is easy to fix in general, either, because of course derivatives in implicit functions can be arbitrarily complicated to solve for, and so not necessarily accessible to a computer method. I don't think there are any numerical methods for doing this.