Definitions:

1. To sharpen or whet;
2. Metaphorically, to sharpen the mind, temper, or courage of someone, to incite, to impel. In the NT it means to provoke or rouse to anger or indignation; only in the middle/passive paroXYnomai.

References:

1. Sept.: Deut. 32:41;
2. Acts 17:16; Sept.: Deut. 1:34; 9:18,19 .

Acts 17:16--While Paul was waiting for [Silas and Timothy] in Athens, he was greatly distressed (paro-XYneto to PNEUma auTOU) to see that the city was full of idols. (NIV)
Alt.: ... his spirit was stirred in him, when he saw the city wholly given to idolatry. (KJV)
Alt.: ... his spirit was provoked within him ... (RSV)
Alt.: ... his spirit within him came to be irritated (NWT)
Alt.: ... he was greatly upset when he noticed ... (TEV)
Alt.: ... he grew exasperated at the sight of idols everywhere in the city. (NAB)

BT Internet:  paroxunomai, be irritable, be greatly upset

'provoked to anger.' "not touchy" Phillips, Barrett.

St. John Chrysostom, Homily 33:  "Is not easily provoked" See love again not only subduing vice, but not even suffering it to arise at all. For he said not, "though provoked, she overcomes," but, "is not provoked."

Adam Clarke:  Is not easily provoked] ou paroxunetai Is not provoked, is not irritated, is not made sour or bitter. How the word "easily" got into our translation it is hard to say; but, however it got in, it is utterly improper, and has nothing in the original to countenance it. By the transcript from my old MS., which certainly contains the first translation ever made in English, we find that the word did not exist there, the conscientious translator rendering it thus: It is not stirid to wrath.

The New Testament, printed in 1547, 4to., the first year of Edward VI, in English and Latin, has simply, is not provokeed to angre. The edition published in English in the following year, 1548, has the same rendering, but the orthography better: is not provoked to anger. The Bible in folio, with notes, published the next year, 1549, by Edmund Becke, preserves nearly the same reading, is not provoketh to anger. The large folio printed by Richard Cardmarden, at Rouen, 1566, has the same reading. The translation made and printed by the command of King James i., fol., 1611, &c. departs from all these, and improperly inserts the word easily, which might have been his majesty's own; and yet this translation was not followed by some subsequent editions; for the 4to. Bible printed at London four years after, 1615, not only retains this original and correct reading, it is not provoked to anger, but has the word love every where in this chapter instead of charity, in which all the preceding versions and editions agree. In short, this is the reading of Coverdale, Matthews, Cranmer, the Geneva, and others; and our own authorized version is the only one which I have seen where this false reading appears.

As to the ancient versions, they all, Vulgate, Syriac, Arabic, AEthiopic, Coptic, and Itala, strictly follow the Greek text; and supply no word that tends to abate the signification of the apostle's ou paroxunetai, is not provoked; nor is there a various reading here in all the numerous MSS. It is of importance to make these observations, because the common version of this place destroys the meaning of the apostle, and makes him speak very improperly. If love is provoked at all; it then ceases to be love; and if it be not easily provoked, this grants, as almost all the commentators say, that in special cases it may be provoked; and this they instance in the case of Paul and Barnabas, Acts xv. 39; but I have sufficiently vindicated this passage in my note on that place, and given at large the meaning of the word paroxunw; and to that place I beg leave to refer the reader.

The apostle's own words in ver. 7, are a sufficient proof that the love of which he speaks can never be provoked. When the man who possesses this love gives way to provocation, he loses the balance of his soul, and grieves the Spirit of God. In that instant he ceases from loving God with all his soul, mind, and strength; and surely if he get embittered against his neighbour, he does not love him as himself. It is generally said that, though a man may feel himself highly irritated against the sin, he may feel tender concern for the sinner. Irritation of any kind is inconsistent with self-government, and consequently with internal peace and communion with God. However favourably we may think of our own state, and however industrious we may be to find out excuses for sallies of passion, &c., still the testimony of God is, Love is not provoked; and if I have not such a love, whatever else I may possess, it profiteth me nothing.

Geneva Notes:  is not easily provoked
It is not insolent, or reproachful.

John Gill's Exposition of the Bible:  is not easily provoked
to wrath, but gives place to it: such an one is provoked at sin, at immorality and idolatry, as Paul's spirit was stirred up or provoked, when he saw the superstition of the city of Athens; and is easily provoked to love and good works, which are entirely agreeable to the nature of charity:

John W. Gregson:  As a spark which falls into the sea hurts not the sea, but is itself extinguished, so an evil thing befalling a loving soul will be extinguished without disquietude or excitement. Love is not provoked, irritable, nor bitter; it does not have a "short fuse."

Yeager says of this verse, "The text does not say that love is never provoked. It does say that the provocation does not lead to a fit of rage."

Matthew Henry:  It tempers and restrains the passions. Ou paroxynetai--is not exasperated. It corrects a sharpness of temper, sweetens and softens the mind, so that it does not suddenly conceive, nor long continue, a vehement passion. Where the fire of love is kept in, the flames of wrath will not easily kindle, nor long keep burning. Charity will never be angry without a cause, and will endeavour to confine the passions within proper limits, that they may not exceed the measure that is just, either in degree or duration. Anger cannot rest in the bosom where love reigns. It is hard to be angry with those we love, but very easy to drop our resentments and be reconciled.

B.W. Johnson:  Is not provoked. Does not fly into a rage, but keeps the temper under control.

Mark Heber Miller:  (Love) does not become provoked.

The Greek is OU PAROXYNETAI and is variously rendered: NEB: not quick to take offense; RSV: not irritable; PME: not touchy; BECK: it doesn’t get angry. One can see the root of "oxygen" in the word and it literally refers to the bellows of the blacksmith which blasts the coals and heats things up and thus sharpens iron. The English word PAROXYSM can mean "a sudden convulsion or outburst" for either good or bad. The word is rare and various forms convey "to stir" or "arouse" (Acts 17:16).

The first case of anger makes us shiver that such a thing befall us. (Genesis 4:5) The Corinthian church had a problem with "cases of anger." (2 Corinthians 12:20) Anger is a work of the flesh. (Galatians 5:20) Elsewhere Paul encourages getting rid of anger. (Ephesians 4:31; Colossians 3:8)

There are two particular occurrences which catch our attention, one bad and another good. Note Acts 15:39, "But, a paroxysm [sharp burst of anger] occurred and they (Paul and Barnabas) separated from one another." It is very interesting that the one who wrote that "love is not (given) to paroxysm" should have it recorded by his traveling companion Luke that he did succumb to such a burst of anger. Though we are not precisely told who it was that first became so angry. The case that caused this circumstance was Paul’s refusal to take the disciple Mark on this missionary tour because he had left midway during the previous journey. Barnabas, Mark’s cousin, may have wanted him to join them for family reasons. (Galatians 4:10) Clearly, here is a case where love was not controlling these men.

This illustrates that there are times when even previously good friends or companions--as was the case with Barnabas and Paul--have a disagreement so severe that they become angry with one another. Paul and his new partner, Silas (Silvanus), go on to write several inspired epistles where Barnabas misses out on this opportunity. Barnabas is not mentioned again in the Book of Acts. However, Paul later mentions both Barnabas (though misled by Judaizers) and Mark in positive tones. (Galatians 2:1, 9, 13; 4:10)

There is a good form of PAROXYSM which occurs in Hebrews 10:24, "incite [PAROXYSMON] to love and fine works." Where love may cool it is vital to blast the coals with the oxygen of encouragement. Paul writes this counsel in the context of Christian meetings.

It is true some personalities are given to wearing their feelings on their cuff and have a low boiling point. This is due more to immaturity on the Christian walk, while those who have been Nazarene disciples longer will manifest a calmer and controlled spirit. It is often easier to learn to remain quiet--and keep opinions within and under control--rather than struggling to always say the right thing. Once one controls rash speech, anger will become less and less part of the Christian character.

There is one final thought regarding love not provoking others. Being a cause of provocation can bring our Christian friends enormous grief. Consider, how Moses was driven "crazy" (Ecclesiastes 7:7) by the provocation of his fellow worshippers. Psalm 106:32, 33 records, "Further, they caused provocation at the waters of Mer'i·bah, so that it went badly with Moses by reason of them. For they embittered his spirit and he began to speak rashly with his lips." (Numbers 20:2, 12; 27:14; Deuteronomy 1:37; 32:51; compare Hebrews 3:15)

Robertson's Word Studies:  {Is not provoked} (ou paroxunetai). Old word. In N.T. only here and #Ac 17:16 which see. Irritation or sharpness of spirit. And yet Paul felt it in Athens (exasperation) and he and Barnabas had paroxusmos (paroxysm) in Antioch (15:39). See good sense of paroxusmos in Heb 10:24.

The Theologian: The Internet Journal for Integrated Theology:  Self-centredness is immensely divisive, and could cause them to be "touchy" when they felt their interests were not being adequately addressed. Love, however, does not fly into paroxysms of rage or make dramatic gestures of discontent at the slightest provocation.

Bill Turner:  Love is not provoked or irritable, it does not fly into a temper.

"Provoked," is "paroxunetai," the present passive of "paroxuno," to sharpen, provoke, or stir up, it only occurs here and Acts.17v16., where it says that Paul's spirit was stirred within him when he saw the city of Athens wholly given to idolatry. There are times when it is a sin not to be angry, but a person who can't control their temper is no use in the service of God, for they can hurt and injure many people. Lk.16v14,15. Mt.23v1-39. 1Tim.3v3. 2Tim.2v24,25. Eph.4v15. Gal.13v26. When the Corinthians misused their spiritual gifts, Paul gave them sound teaching and advice, not an exhibition of bad temper.

Love never gets angry and forbids spiritual gifts, nor does it call this bad temper and unbelief by the name of righteous indignation. For the noun "paroxusmos," see Acts.15v39, where Paul and Barnabas both failed to manifest "agape" love and had a most unchristian angry dispute. See Heb.10v24, where "paroxusmos" is used in a good sense, "And let us thoughtfully consider one another to provoke unto love and to good works." Here Paul uses "katanoomen," the present active subjunctive of "katanoeo," to put the mind down upon, to thoughtfully consider; and he uses the present tense to show that "agape" love always thoughtfully considers how to provoke to love and good works, and is never provoked to fly into a temper.

Vincent's Word Studies:  Easily provoked (paroxunetai). Easily is superfluous and gives a wrong coloring to the statement, which is absolute: is not provoked or exasperated. The verb occurs only here and Acts xvii. 16. The kindred noun paroxusmov, in Acts xv. 39, describes the irritation which arose between Paul and Barnabas. In Heb. x. 24, stimulating to good works. It is used of provoking God, Deut. ix. 8; Psalm cv. 29; Isa. lxv. 3.

Acts 15:38--Paul did not think it wise to take [Mark], because he had deserted them in Pamphylia and had not continued with them in the work. 39 They had such a sharp disagreement (paroxysMOS) that they parted company. Barnabas took Mark and sailed for Cyprus (NIV)
Alt.: And there arose a sharp contention (RSV)
Alt.: There was a sharp argument, and they separated (TEV)
Alt.: There was a sharp clash of opinion, so much so that they went their separate ways (Phillips)

Heb. 10:24--And we should bear each other in mind (katanoO-men alLE-lous), for an incitement (paroxysMON) of love (aGApe-s) and good works (kaLO-N ERgo-n). (ED)
Alt.: And let us consider how we may spur one another on toward love and good deeds. (NIV)
Alt.: ... how to stir up one another to (RSV)
Alt.: And let us consider one another to provoke unto (KJV)
Alt.: Let us be concerned for one another, to help one another to show love and to do good. (TEV)
Alt.: and let us think of one another and how we can encourage one another to love and do good deeds. (Phillips)
Alt.: We must consider how to rouse one another to (NAB)