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In "Patients and doctors" (Kenneth Walker), it says
The great advantage of taking medicine is that it makes no demands on the taker beyond that of putting up for a moment with a disgusting taste, and that is what all patients demand of their doctors--to be cured at no inconvenience to themselves.
I think that "demand" in "demand of their doctors" is a verb, instead of a noun, am i right?
You are right to say "demand" is a verb. In this case the subject is "all patients". If "demand" were to be a noun:
The demand for cars has been rising
...that is what all patients demand of their doctors...
I add to what @BillJ has rightly commented, that demand in OP's example is a verb.
To use the noun demands using OP's example, we need a few modifications.
As the verb would no longer be there in the what-clause, we need to add one, like are.
...that is what all patients' demands of their doctors are....
...that is what all patients' demands are of their doctors....
Please note that I have used the plural demands, as @FumbleFingers has rightly said, and the possessive patients'.
demands [ plural ] the difficult things that you have to do: The demands of nursing are too great for a lot of people. Cambridge Dictionary
