I have numerous gears, axis and other elements from different Lego Technic sets. I wonder if using them I can make a kind of a differential like this?

this

If not, how can I implement similar functionality in case I have two motorized wheels on one axis that can be steered and connected to motor(s) via cardans like this:

enter image description here

This can definitely be accomplished, although it will be difficult to create a differential as small as the dedicated piece that you mentioned.

Some builders create their own differentials for use in larger models because of the inherent torque limits and slippage that the traditional differential and single bevel gears introduce. Here's one example (source) of a differential that can be created using fairly common elements:

heavy duty differential

The key is to build the frame containing the three gears and then attach a fixed gear to the frame. The tricky part is finding a gear that can pass an axle through the center without being fixed on it. In the example above, half of a turntable was used, but there are a few other parts that could work as well. Here's the turntable (48452):

turntable 48452

If you are comfortable with modifying your parts, you could also drill out the center of a gear to allow it to turn freely.

The delicate problem in a brick-built differential is to create the frame. It must have a gear to drive the differential and this gear must let the axle rotate freely. Without parts modification, the only possibility is to use a turntable, either the big one or the small one. Anyway, the solution is bulky and not economical (the differential parts are cheap!). The big version can be interesting nonetheless in large vehicules that require a lot of torque, see here.

Yes - in fact, you can even improve upon the basic design, to create a limited-slip differential.

Here's one by Rob Stehlik, via http://www.technicbricks.com/2009/03/tbs-techtips-22-torsen-differential.html

enter image description here

The key here, which other answers don't note, is that the rotating part of the axle does not have to pass through the center of the cog. Instead, here it is offset through the outer holes.

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