Calculus with polynomials
In
mathematics, polynomials are perhaps the simplest
functions with which to do
calculus. Their derivatives and integrals are given by the following rules:


Hence the derivative of x
^{100} is 100x
^{99} and the integral of x
^{100} is x
^{101}/101 + c.
Because differentiation is linear, we have:
So it remains to find for any
natural number r. The derivative of function f(x) is given by Newton's difference quotient
By the
binomial theorem, and using the Cnotation of
combinations,
and therefore
The derivative is the limit of this as
which gives the claimed result.
Generalisation

is generally true for all values of k where x^{k} is meaningful. In particular it holds for all rational k for values of x where x^{k} is defined.
Similarly for integration, see table of integrals.