Newton’s method boils down to replacing your function by a first-order approximation. For a differentiable function, in a small neighbourhood(!), that’s a good approximation (by definition), though, and the zero of the model function will be very close to the zero of the original function (if it lies in that neighbourhood).

PS: i did not expect the poster and author to be the same person, otherwise I would’ve phrased my criticism differently. A SHOW HN would have helped.

PPS: basically the whole reciprocal/inverse confusion only arises because you start the multidimensional case from your iteration formula. If you back to its derivation, and start again from there, you can avoid that.