The easiest way is just grasp the slide with the off-side hand, pull back and release. Not only does this save wear on the thumb, but not very many years ago was taught as the correct way to drop the slide.
FYI, and contrary to un-informed current use, the part you refer to is the slide STOP. Its primary function is to hold the slide open. When disengeged it releases the slide, but thyat is really incidental to its primary operation.
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Gunsmithing since 1961
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