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I'm aware that actual focal length of a lens on APS-C cameras is longer than what is written on the lens.

My question is, for lenses that are specially designed for APS-C cameras (like a Nikkor DX lens), do they write actual focal length on the lens or it is the focal length if mounted on a full-frame camera?

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I'm aware that actual focal length of a lens on APS-C cameras is longer than what is written on the lens.

Sorry to say this, but whoever told you that was wrong. The actual focal length of a lens on an APS-C camera is exactly1 what is written on the lens. What you're probably confused about is the fact that the field of view you get from mounting a 50mm lens on an APS-C camera is the same is that of mounting a 75-80mm lens on a full-frame camera (this varies a bit because "APS-C" sized sensors vary a bit between manufacturers). There's a lot of good information about this in our existing question at Does my crop sensor camera actually turn my lenses into a longer focal length?

  1. Within normal manufacturing tolerances, manufacturers rounding numbers, etc. But that applies equally to all lenses.
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The "crop factor" of sensors smaller than full frame is never respected in the specs of lenses. A 50mm EF-S lens has the same focal length as a 50mm EF lens, and will produce the same field-of-view on the same camera.

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