I am trying to understand this formula to find the transconductance of a MOSFET transistor, but I am not getting the passage from the second to the third line. What happened there?
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The basis is the following approximate equation for the drain current (Wikipedia):
Taking the partial derivative gave gm as a function of (Vgs-Vth). One can express (Vgs-Vth) as a function of Id reversing the base equation. By inserting that one gets your problematic third line.