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Can you clarify how this term changes depending on the direction we move? I'm not sure I fully understand how this approximation works


What is the relationship between the gradient and the tangent plane?


Why can we consider each coordinate independently when calculating direction of greatest change?


it's clearly that this approximation has a limit: x cant be too far from x0. in practice is there a way to define the range x we can use to approximate? or is there a way to know if we have done too far?