Car's purpose -- at root to do with self, in the first place (10)

I believe the answer is:

automotive

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'root to do with self in the first place' is the definition.
I can't tell whether this definition defines the answer.

'car's purpose' is the wordplay.
'car' becomes 'auto' ('auto' can be a synonym of 'car').
'purpose' becomes 'motive' (I've seen this before).
'auto'+'motive'='AUTOMOTIVE'

'at' acts as a link.

Can you help me to learn more?

(Another definition for automotive that I've seen is " Relating to motor vehicles".)

I've seen this clue in The Telegraph.

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