How is the differential acceleration in a gravitational field derived using Taylor expansion

Mathematical objects: General Derivation.

Question: How is the differential acceleration in a gravitational field derived using Taylor expansion?

Answer: The differential acceleration is derived by performing a Taylor expansion of the gravitational field around the position, retaining only the first-order terms in the small displacement.

Proof and Derivation: Derivation and Calculation of the Gravitational Tidal Tensor.

Audio Overview: Black Holes to Ocean Tides: Decoding the Gravitational Tidal Tensor