# Calculus with polynomials

 related topics {math, number, function}

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 x100 is 100x99 and the indefinite integral of x100 is $\frac{x^{101}}{101}+C$ 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

### Power rule

The power rule for differentiation states that for every natural number n, the derivative of $f(x)=x^n \!$ is $f'(x)=nx^{n-1},\!$ 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 right-hand side.

### Proof

To prove the power rule for differentiation, we use the definition of the derivative as a limit:

Substituting f(x) = xn gives

Since the numerator is a difference of powers, it factors:

We can simplify the first factor:

Then we can cancel the h:

Once the h has been canceled, we may evaluate the limit by substituting h=0.

Each summand is now the same xn − 1 so: