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I think the question is clear from the title.

How the weight function for example in Huber is calculated?

Is it by differentiating the objective function?

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jeza
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    Where's this table from? Maronna, Martin and Yohai "Robust Statistics: Theory and Practice" defines the weight as $\psi(x)/x$, $\psi$ probably what you call "score function". This doesn't correspond 100% to your table, but then at least the bisquare score and weight look different in that book, so the source of your table may use different definitions. Generally in mathematics any notation and terminology is not generally fixed but what the author defines it to be, which we'd need to know to make sense of that table. – Christian Hennig Nov 16 '20 at 23:20
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    I am talking in general, not in specific sources. – jeza Nov 17 '20 at 11:07

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