Below 2 months of age, respiratory rate
more than 60 per min is abnormal.
To have the correct rate, count the respiration when baby is asleep.
Does he have fever? How is the feeding? Does he have difficulty while feeding? Fever with cough
and fast breathing suggests pneumonia
that is treated with antibiotics usually.
After treatment with antibiotics is over, cough alone can persist for few more days.
If he still has all the symptoms and its almost 10 days now, its wise to have x ray
chest done before starting another antibiotic.
So have him examined again, look for feeding difficulties and if its just a cough/ cold
and no other problems, and his chest is clear, he might not need another course of antibiotics.
Hope my answer helps you.
Wish a speedy recovery for your kid.