Well, by the addition property of differentiation we know that the derivative of a sum is the sum of the derviatives . So, when we differentiate a polynomial we know that all we do is differentiate each term of the polynomial and take the sum. Furthermore, the power rule states that when differentiating a term in a polynomial, all we have to do is the following:

Therefore, a polynomial of the form when diferrenyiated becomes , and it is seen that in each successive derivative, the leading coeficient is reduced by 1.

A prrof of the power rule and addition property can be found here:

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