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Is the depth of field really different when using cameras with different size of their sensors? You should think it is literally just a cropped image. How could that possibly affect the sharpness?

The image left shows the ratio of the size of a crop (here canon APS-C size) compared to so called Full Frame size bodies. 

With some rather simple experiment I just checked how the sensor's size influences the depth of field. I photographed some standard ruler with two different bodies and with different focal length so that both images are taken with roughly the same angle. 

This one is taken with an EOS 10D, a Canon crop, and the classic EF 50mm 1.8 with an aperture of f/5.6. 

You see some sharp area in the middle, between roughly 17 and 20 centimeters, and some increasing blurr the farer your eye moves away from this focal distance - in both directions.  

This is significantly different on the following image which was taken with a Canon EOS 5D, a full frame body, with 5.6. For this one I used the 85mm 1.8, which short of matches equivalent of 50mm on crop. (crop factor is 1.6, therefore roughly 1.5).

Click on the images and enlarge them and compare the original sizes. You will notice that the FF of the 5D will quicker turn blurry. Don't be mistaken by the fact that the overall sharpness of the 85mm lens is lightly better than the 50mm. It gets clearer the more away the spots are from the focal distance that you compare. 

It gets even more impressive on the following image where I scaled both images and mounted the rulers next to each other so that you can compare them in a glance.