Cleaner of busy royal fellow (9)

I believe the answer is:

detergent

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'cleaner' is the definition.
(chemical that cleans)

'of busy royal fellow' is the wordplay.
'of busy' becomes 'det' (I can't explain this - if you can you should give a lot more credence to this answer).
'royal' becomes 'ER' (abbreviation for Elizabeth Regina).
'fellow' becomes 'gent' (synonyms).
'det'+'er'+'gent'='DETERGENT'

Can you help me to learn more?

(Other definitions for detergent that I've seen before include "Cleaning product" , "Soapless cleanser" , "Get tender (anag)" , "Cleansing substance used for washing clothes" , "Cleansing solvent" .)

I've seen this clue in The Times.

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