14th  Differential_calculus">Top calculus topics: Differential calculus 
In mathematics, polynomials are perhaps the simplest functions used in calculus. Their derivatives and indefinite integrals are given by the following rules:
and
Hence, the derivative of x^{100} is 100x^{99} and the indefinite integral of x^{100} is where C is an arbitrary constant of integration.
This article will state and prove the power rule for differentiation, and then use it to prove these two formulas.
Contents 
The power rule for differentiation states that for every natural number n, the derivative of is that is,
The power rule for integration
for natural n is then an easy consequence. One just needs to take the derivative of this equality and use the power rule and linearity of differentiation on the righthand side.
To prove the power rule for differentiation, we use the definition of the derivative as a limit:
Substituting f(x) = x^{n} gives
One can then express (x + h)^{n} by applying the binomial theorem to obtain
The i = n term of the sum can then be written independently of the sum to yield
Cancelling the x^{n} terms one generates
An h can be factored out from each term in the sum. From thence we can cancel the h in the denominator to obtain
To evaluate this limit we observe that n − i − 1 > 0 for all i < n − 1 and equal to zero for Thus only the h^{0} term will survive with i = n − 1 yielding
Evaluating the binomial coefficient gives
It follows that
To differentiate arbitrary polynomials, one can use the linearity property of the differential operator to obtain:
Using the linearity of integration and the power rule for integration, one shows in the same way that
One can prove that the power rule is valid for any real exponent, that is
for any real number a as long as x is in the domain of the functions on the left and right hand sides. Using this formula, together with
one can differentiate and integrate linear combinations of powers of x which are not necessarily polynomials.
