Why is the moon's gravity differential field pointing away from the

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You have to understand what is meant by a differential field. Here, it's the difference between the gravitational field generated by the moon at that point and the field it generates at the Earth's centre. Since the moon's gravity is weaker further from the moon, the field it generates towards itself there is less than its field at the Earth's centre, so the difference points away from the moon.
More subtle is working out that when the moon is on the horizon the differential field points towards the Earth's centre.