In much of the “common literature”, it is customary to define an “integer polynomial derivative” using the same conventions as the usual sense of “derivative”. Just like with the customary definition of derivative/anti-derivative, there are other options.
The definition I will use is one I came up with at age 10. Yes, it is a very simple definition.
Let the “pseudo-derivative” of an integer polynomial p (i.e., a polynomial with integer coefficients and supplied with an integer variable) be defined as p(x+1) – p(x). Let p`(x) (read “p back-tick of x”) be the representation of the pseudo-derivative of p(x), i.e., p`(x) = p(x+1)-p(x).
This definition allows for certain features that the traditional derivative cannot claim, at least not without modifications beyond this scope.
In particular, when p`(x) is used, it is possible to construct the value of p(x) for any given x using only a sum based on the pseudo-derivatives of p(x) up to the point where p(x)^[n] = 0 (under the convention that x^[n] is the nth pseudo-derivative of x).
Under this definition of pseudo-derivative, many of the same rules apply as to the proper derivative, such as constant multiples pass through unchanged, pseudo-derivative of a sum is the sum of the pseudo-derivatives, however the product rule is no longer effective. Here it is for proper derivatives:
d/dx(p(x)q(x)) = p'(x)q(x) + p(x)q'(x) + p'(x)q'(x)
Under the usual definition, the third term becomes 0 in the limit that is applied, but here there is no limit.
Other rules are probably different as well, let the reader make note of any that are of interest.
Here’s an example of how this derivative works out:
p(x) = ax^3 + bx^2 + cx +d
x | p(x) | ——————– p(x+1) – p(x) | p`(x+1) – p`(x) | p“(x+1) – p“(x)
0 – d
——————————| a + b + c
1 – a + b + c + d ———————————-| 6a + 2b
——————————| 7a + 3b + c ———————| 6a
2 – 8a + 4b + 2c + d —————————–| 12a + 2b
——————————| 19a + 5b + c ——————-| 6a
3 – 27a + 9b + 3c + d —————————-| 18a + 2b
——————————| 37a + 7b + c
4 – 64a + 16b + 4c + d
The above pseudo-derivative tree applies to any cubic equation; equations of higher degree will have similar trees. Note that the final column (the nth derivative of an n-degree polynomial) will always be constant, and will have the value a*n!, where a is the coefficient of the highest-degree term.
Using this tree, it is possible to construct any later value in it based only upon the first term in each pseudo-derivative column and the value of p(0). Simply add the next constant term (in this case, 6a), then add that to the last value in the column next to it (24a + 2b), then repeat the process back through the columns and down in rows until you reach your desired value.