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# Calculus with polynomials

Polynomials are perhaps the simplest functions to do calculus with. Their derivatives and integrals are given by the following rules:

Hence the derivative of x100 is 100x99 and the integral of x100 is x101/101 + c.

## Proof

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 C-notation 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 xk is meaningful. In particular it holds for all rational k for values of x where xk is defined.

## References

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