am i doing the push ups wrong?

usually my triceps is first to go and cant go on

I do 4-6 sets of 15 push ups a day but last 2 sets are very difficult

should I feel more burn in my chest or is it normal for triceps to fatigue first?

I understand that I might be out of shape, I have not work out in about 4 years but when I did I was in pretty damn good shape

I'm 29 years old 230lbs

any advice