Question

Much of our success___(depend/depends)on our efforts.

Answer 1

Whether to use "Depend" or "Depends",  depends on the fact that we need to see the word on which "Depend/Depends" is acting upon.
In this case, it needs to apply to "efforts", which is in plural.
Hence the answer is:
Much of our success <depends> on our efforts.

If the sentence was something like:
Much of your success will <depend> on this interview. 
then <depend> is acting upon "interview", hence in singular.

Answer 2

Answer:

Depends

Explanation:

Much of our success Depends on our efforts.