In the most basic terms, it really doesn't matter if a lens is FX/full frame capable or DX/crop only design. In most cases you won't find the same focal lengths in both. But, for example, Nikon now makes 70-300mm DX and 70-300mm FX lenses. That's unusual and the lenses in this case use different optical formulae, so will have different performance in many other respects. But focal length is focal length, regardless of camera sensor size. The above 70-300mm are both still 70-300mm. Or a theoretical 50mm DX lens and 50mm FX lens would both still be 50mm lenses.

What differs is how the camera uses the focal length.... How any given focal length "behaves" on the different sensor formats.

In fact, this is nothing new or unique to digital imaging. The same thing happened with film cameras. In fact, many digital sensor formats are named after film formats.... "full frame" refers to 24x36mm images produced by many 35mm wide roll film cameras. "APS-C" and "APS-H" refer to a short-lived film format that Kodak created (as was the case with many other formats). Even modern "medium format" digital are based upon cameras that use 70mm wide roll film.

A 50mm lens on a "large format" film camera (4x5" or approx 100x125mm) "acts as" an ultrawide.
That 50mm lens on a medium format film camera (6x6 or 6x7cm) "acts like" a wide lens.
A 50mm lens on a 35mm film or "full frame" digital camera behaves as a "normal" or "standard" lens.
The same 50mm lens on an APS-C digital will act as a short telephoto.
Or that same 50mm lens on a super small digital sensor such as a 1/2.3" will become a powerful telephoto.

Now, that 50mm has to be designed differently for some of the many different formats... It would need different diameter elements and optical formula to make a much larger image circle to cover large format film, for example, than is needed for 35mm/full frame or APS-C. And, depending upon format it will often need to focus the image and as best possible align various colors of the spectrum to all come together at a particular distance behind the lens. This "back focus" dimension has to vary too, depending upon format.

But it's still a 50mm lens! FOCAL LENGTH DOESN'T CHANGE. What changes is the how the camera and its particular sensor or film format USES that focal length.

So... why choose an FX full frame or a DX/crop-only lens? Well, in some cases you simply don't don't have a choice. Most telephotos are FX designs, for example. There's really not much need to design many "crop only" teles. Alternatively, if you want a truly wide lens to use on a DX camera you 'll often be best served with a DX lens. An FX/full frame lens with equivalent angle of view on a DX camera will necessarily be bigger, heavier and probably a whole lot more expensive. For example, compare a Canon EF 11-24mm f/4L USM full frame-capable lens with a Tokina AT-X 11-20mm f/2.8 DX/crop only lens.... much the same range of focal lengths, but vastly different in size, weight and cost!

Or, turn it around and ask why one might choose a DX/crop camera over an FX/full frame model or vice versa? Well, someone who uses telephotos a lot for sports or wildlife might prefer a camera that give all the quality one needs with a hand-holdable 3 lb., $1500 300mm f/4 lens... Instead of needing to spend $9000 or more to get similar angle of view on an FX/full frame camera with a 7 or 8 lb. 500mm f/4 lens that's going to need a sturdy tripod or at least a monopod for anything longer than a few minutes shooting. Conversely, someone who shoots a lot of scenics might prefer an FF/full frame camera for it's "bigger" images, potential to make larger prints, and how it works with wide angle lenses.

You're also correct that Depth of Field doesn't change just because of a different sensor size.... all other things being equal. HOWEVER, other things never remain equal.

In the most basic terms, DoF only changes due to focal length, aperture and distance to the object. If one is using a particular focal length on an APS-C camera, then switches to using a full frame camera... in order to frame the subject the same way you will need to either move closer or use a longer focal length. And either of those actions will cause DoF to be rendered more shallow.

The reason DoF appears to change with different formats on Depth of Field calculators is because they're defining DoF differently, depending upon format. Notice how the earlier post screen captures of APS-C versus full frame DoF calculations use .02mm and .03mm "circle of confusion". This accounts for most of the differences those calculators show... but I think is just confusing (). It's more relevant that we also need to change distance or focal length or a bit of both, whenever we change sensor format, in order to frame the subject the same way.

There's also diffraction to take into account. That's an optical effect causing loss of fine detail in images when "too small" a lens aperture is used, such as might be done in an effort to increase DoF. Diffraction is "less" with larger full frame or medium format sensors, than it is with smaller sensor formats such as APS-C or m4/3. But the difference in diffraction actually mostly has to do with difference in magnification of any given image when it's prepared for it's intended use. For example, to make an 8x12" print from a full frame image requires approx. 8X magnification. In comparison, an APS-C image printed to the same size would need about 13X magnification. And, the more magnification there is, the more obvious any loss of fine detail might be.

So, even though DoF doesn't actually "change" with different formats.... in real world usage of cameras, lenses and the images made with them... compared to APS-C or m4/3 a full frame sensor camera will SEEM to both render stronger blur effects at larger apertures AND to be more tolerant of small apertures when greater DoF is wanted.

It's actually a lot more complex and there are far more variables.... but IMO these are the main points.

P.S. The "naming" of formats is actually sort of funny... At one time "full frame" was considered "miniature", back when most cameras were using film that was twice as wide or came in individual sheets! "APS-C" is even more of a hoot... "APS" stands for Advanced Photo System film format that Kodak created, primarily for more amateur oriented, highly automated point-n-shoot cameras.... but their timing really sucked! Using 24mm wide film (50% more efficient than 35mm film), APS was introduced in 1996... when digital was just getting started. WIthin five years digital point n shoots had virtually killed off the film cameras targeting that same market. In 2004, after only 8 years, Kodak retired the system. I don't know if anyone is still making APS film cartridges and pretty few places can process it now.

Simply put on a given body it doesn't matter if a lens is dx or fx the dof will be the same.

If you change camera bodies to one with a different pixel size the dof will be different to the first camera but not different between the fx and dx lens. Its the sensor difference that makes the change not the lens.

Perhaps the more important take is you can't get quite the same results with the same settings with different camera models.
Can you see the difference, with my eyes, not really even wearing my glasses :)

I guess that makes camera's have a unique viewpoint, so a D750 will not see a scene exactly the same as a D850 even with the same lens mounted. Does it matter, does one or the other have an edge in making beautiful photo's?

Maybe the Leica folks have a point with the "Leica look".

A FX lens has the same field of view (angle) as a DX lens but when creating the Circle of confusion the angle is narrower out of the lens on a DX lens when compared to a FX lens as both use the same distance lens/sensor. The result is a change of behavior when calculating the DoF. This is what I understood when looking at it.

You can see the change using your calculator. Select a lens length, the aperture then switch from a Dx to a FX camera.

Three captures from your site.

FX DoF are the same regardless of pixel density.
DX DoF is different. Note the change of DoF distribution.

So something is going on and I believe this is due to the lens compressing the circle of diffusion. I have no proof of that but it is the only reasonable explanation I have. It does not mean I am right assuming this.

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A FX lens has the same field of view (angle) as a DX lens but when creating the Circle of confusion the angle is narrower out of the lens on a DX lens when compared to a FX lens as both use the same distance lens/sensor. The result is a change of behavior when calculating the DoF. This is what I understood when looking at it.

You can see the change using your calculator. Select a lens length, the aperture then switch from a Dx to a FX camera.

Three captures from your site.

FX DoF are the same regardless of pixel density.
DX DoF is different. Note the change of DoF distribution.

So something is going on and I believe this is due to the lens elements compressing the circle of diffusion. I have no proof of that but it is the only reasonable explanation I have. It does not mean I am right assuming this.

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RGG, there is no Dof calculator that supports your claim.

I posted the results from DoF Master, but I could have used any one of a dozen different ones, and the result is the same - DX has less DoF than FX with the same lens, same distance, same aperture. I will post again.

In this illustration - 50mm, F8, 10 ft subject to focal plane distance - the only thing that changes is the camera body.

The FX camera has a total DoF of 6.28 ft and a hyperfocal distance of 34.3 ft, and the DX camera has a total DoF of 3.98 ft and an HD of 51.4. Everything is the same except for the smaller sensor. What am I missing here?