【Active】 , 【operative】 , 【dynamic】 and 【live】 are synonymous when they mean being at work or in effective action.

【Active】 in general may be employed wherever the others are applicable, but it is also usable where none of the others would be appropriate. It may qualify anything that shows its nature or its existence in acts, in action, or in work.

• an 【active】 volcano
• an 【active】 brain
• 【active】 sympathy

And it is applicable to anything which can be worked, operated, manipulated, or wielded.

• an 【active】 pen
• a mine still 【active】 after fifty years of mining

It is also applicable to an agent, an operator, an instrument, a means or to something accomplished by any such agency.

• 【active】 enforcement of the law
• an 【active】 propagandist
• an 【active】 search for truth

【Active】 may imply little more action or movement or exertion than shown in a state that is not death, rest, or inertness.

• his pulse is low, but his heart is still 【active】

Oor it may, and usually does, imply vigor and energy in action or movement.

• an 【active】 market
• an 【active】 writer
• the 【active】 stage of a disease

Often it suggests causation or activation.

• the 【active】 principle in a drug

In contrast with 【active】 【operative】 is applicable only to things (as a principle, motive, emotion) that have a capacity for acting, working, or effecting ends or to those (as laws) that can be put into operation.

• when strict ethical principles are 【operative】 in society, men may expect the millennium
• the rule has been 【operative】 since January first

【Operative】 usually is weaker in its implication of effectiveness; one’s sense of duty is 【operative】 when it in any degree influences one’s thoughts or actions; it is 【active】 when it serves as a spur to action or is the determinant of one’s actions.

【Dynamic】 stresses the realization of the potential in something: it therefore often connotes release of great energy and consequent forcefulness; a 【dynamic】 personality is one that exhibits great power and exerts a great influence; love is a 【dynamic】 emotion when it sweeps away all that would obstruct its movement.

【Live】 is also applicable to persons, personalities, principles, laws, emotions, and motives. It stresses vitality and modernity more than forcibleness and, when used of persons, intelligent awareness of present conditions or needs and progressiveness more than effectiveness.

• Steve had in him the making of a 【live】 man of affairs
  —Anderson