Calculus with polynomials

related topics
{math, number, function}

Power rule, Product rule, Quotient rule, Chain rule

Lists of integrals
Improper integrals
Integration by:
parts, disks, cylindrical
shells
, substitution,
trigonometric substitution,
partial fractions, changing order

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:

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