Jerking in automatic transmission - what is the reason and how to fix it?

My problem is related to the automatic transmission. When shifting to D the car jerks. At 60 km/h I also feel a jerk. Other than these two problems, the transmission works flawlessly. Please give me your opinion on this topic.

Jerking in automatic transmission - what is the reason and how to fix it?
2001_BMW_L7_(E38)_sedan_(2015-02-13)_02

BMW driver question

My problem is related to the automatic transmission. When shifting to D the car jerks. At 60 km/h I also feel a jerk. Other than these two problems, the transmission works flawlessly. I've been to the mechanic and he told me that there is play on the half axle and on the flexible link between the shaft and the rear bridge. Please give me your opinion on this subject.

The problem affected a BMW E38 740i model, but it also happens in other models.

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BMW E38 | By Pitexpl (Own work)[CC BY-SA 4.0], via Wikimedia Commons

Response from BM Cars service

In cars with automatic transmissions, the moment of activation of the Drive gear or reverse gear is always combined with a gentle jerk. This must not cause any deafening banging type sounds (loose transmission components) or cause the vehicle to react violently. In addition to checking all transmission components , we also suggest performing a full transmission diagnostic.

We invite you to take advantage of our service.

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BMW E39 | By OSX (Own work) [Public domain], via Wikimedia Commons
szarpanie automatycznej skrzyni BMW

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BMW automatic transmission lever | By Anslaton [CC BY 3.0]