Question

what is the difference between integration and deffiration of any vector like |A|-|B|​

Answer 1

Differentiation means to divide the function in infinitesimally small parts to get the rate of change of one variable with respect to the other or the tangent at a point.

Integration means to add up the infinitesimally small parts of the function or the tangent locuses of the function from the rate of change to get the original function.

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To take the derivative of a vector-valued function, take the derivative of each component. If you interpret the initial function as giving the position of a particle as a function of time, the derivative gives the velocity vector of that particle as a function of time.

Differentiation represents the rate of change of a function. Integration represents an accumulation or sum of a function over a range. They are kind of inverses of each other.