I have a bicycle with a Shimano Altus 2x9 drivetrain. It has 36 and 22 tooth chainrings and a 11-34 tooth cassette.
I am hoping to upgrade this to extend its high range (i.e. 36/11) without sacrificing the low range (22/34), and keeping it 2x. Is this possible at all? I understand that product recommendations are not allowed here, but I am not asking "what is the best". I am asking if there exists a solution to my problem at all (perhaps I am trying to get too much range from a 2x?), and how I might go about finding it and researching compatibility.
I am not experienced with bicycle technology, and I became quite lost when trying to understand what the compatibility constraints are, and what can be mixed with what. For example, I found out that chainrings cannot always be swapped freely, and that not every small / large chainring combination is possible due to pin placement. It is also unclear what are the limits on largest rear cog / large front chainring combinations, what the relevance of the chainline numbers are, etc.
To make this question more concrete, here's an example of trying to find a solution:
Ignoring compatibility, I can do the following quick calculation: I am looking for at least a 10% improvement. At the rear, I seem to be limited to a 11 tooth cog, which means that I need a larger large chainring, with at least 40 teeth to get the 10%. I see that Shimano has 11-42 tooth cassettes for 2x10 systems. In order not to lose the low range, I would need 42/34*22 = 27 tooth chainring. Let's say 28, as that would bring minimal loss.
After some googling, it seems that 40/28 cranksets exists for 2x10 systems in the form of FC-M785, although it seems to be an old model with low availability and high price. It is unclear if it is possible to just change the chainrings on my current Altus crankset (chainrings seem more available). Recently, I learned about the concept of "capacity". For the combination I propose, it seems I would need a capacity of 40-28 + 42-11 = 43. However, the 2x10 rear derailleurs on Shimano's website (RD-M5120-SGS and RD-M4120-SGS) seem to have a capacity of only 41 teeth.
At the moment I am "running out of gears" and want to pedal faster than the 36/11 combo permits. On a good day I am pedaling continuously in top gear, and I would like to be able to go faster.