न + जायते + मृयते + वा + कदाचित् = न जायतेमृयते वा कदाचित्
na + jāyate + mṛyate + vā + kadācit = na jāyate mṛyate vā kadācit
No sandhi at all applies to any of these words yet, because they are all words ending in a vowel, followed by a consonant.
न + अयम + भूत्वा + भाविता + वा + न + भूयः = नायंभूत्वाभावुता वा न भूयः
na + ayam + bhūtvā + bhāvitā + vā + na + bhūyaḥ = nāyaṁ bhūtvā bhāvitā vā na bhūyaḥ
The “na” at the beginning of the line will change unvoiced “t” to voiced “d” at the end of kadācit from the end of the previous line. This is because in sankrit only the midway point of the śloka is really a “new line.” The rule working there is: “Stops without voice are like me and you. We get what we lack when we meet a sādhu.”
“Na + ayam” becomes nāyam, on the “A makes everyone stronger” rule.
Nāyam + bhūtvā becomes nāyaṁ bhūtva, on the rule: “Simple M becomes a dot, way up in the air.”
There is no change for the rest of the words, for they all end in vowels, followed by consonants.
अजः + नित्यः + शाश्वतः + अयम + पुराणः = अजोनित्यः शाश्वतो ‘यं पुराणः
ajaḥ + nityaḥ + śāśvataḥ + ayam + purāṇaḥ = ajo ‘nityaḥ śāśvato ‘yaṁ purāṇaḥ
Ajaḥ + nityaḥ = ajo nityaḥ, following the rule: ““When S (Ḥ) comes with A, there’s a few things to test. Is that consonant voiced? Then make A O and drop S(Ḥ).” Nasals are considered voiced sounds.
Nityaḥ + śāśvataḥ has no change. It doesn’t follow the above rule because sibilants (ś in this case) are not voiced sounds.
Śāśvataḥ + ayaṁ = śāśvato ‘yam. If follows the “make A O and drop Ḥ” rule (as above), and since the voiced sound is “a”, “A too disappears.”
‘yam + purānaḥ = ‘yaṁ purānaḥ, following the rule, “Simple M becomes a dot, way up in the air.”
na hanyate hanyamāne śarīre
There is no sandhi on this line at all, because all the words end in vowels, followed by consonants. However, the “na” at the beginning of the line causes purāṇaḥ at the end of the previous line to become purāno – following the “make A O and drop Ḥ” rule, since nasals are voiced.