3.3. DERIVATIVES245In the first term above, the first factor is fixed and the second goes to zeroas△t→0, while in the second term, the first factor is bounded (since△−→x/△tconverges to−→v) and the second goes to zero. Thus, the wholemess goes to zero, proving that the affine function inside the absolute valuein the numerator on the left above represents the linearization of thecomposition, as required.An important aspect of Proposition3.3.6(perhapstheimportant aspect)is that the rate of change of a function applied to a moving point dependsonlyon the gradient of the function and the velocity of the moving pointat the given moment,noton how the motion might be accelerating, etc.For example, consider the distance from a moving point−→p(t) to the point(1,2): the distance from (x,y) to (1,2) is given byf(x,y) =radicalbig(x−1)2+ (y−2)2with gradient−→
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