f(x+h)-f(x)

f'(x) = lim -----------

h-> 0 h

For example:

f(x) = x^n

f'(x) = n*x^(n-1)

Something similar can be defined for sequences, in that case the 'derative' is called the difference sequence.

a_n = {1, 16, 81, 256, ... }

f(n) = n^4

f'(n) = 1 + 4n + 6n^2 + 4n^3

f''(n) = 14 + 24n + 12n^2

f(3)(n) = 36 + 24n

f(4)(n) = 24

For an arbitrary sequence f:

f(m)(n) = Sum[(-1)^(k)*Binomial[m,k]*f[n-k+m],{k,0,m}]

Here also we see that Pascal's Triangle has a crucial meaning.

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