Yes you need to provide medical of all non accompanying dependents.
You husband would need to submit proofs of divorce.
There might be some terms agreed in divorce about the custody of the child that may prove that he is unable to provide the same.
In my case, my divorce certificate had those points mentioned that i will no longer have any rights or custody over my child.
This along with a letter of explanation helped them understand my situation.
In reply they sent me a form stating that as per R117(9)(d) Humanitarian grounds I will never be able to sponsor my child for immigration in future.
You need not worry, this is a standard procedure. All you have to provide is evidence.
Note: You may also use emails or mail receipts to show that he tried to communicate with ex wife but there was no response. In some cases it was found that ex spouse replied stating that they will not cooperate and that was evidence enough.