## FANDOM

225 Pages

In calculus, the power rule is one of the most important differentiation rules. Since differentiation is linear, polynomials can be differentiated using this rule.

$\frac{d}{dx} x^n = nx^{n-1} , \qquad n \neq 0.$

The power rule holds for all powers except for the constant value $x^0$ which is covered by the constant rule. The derivative is just $0$ rather than $0 \cdot x^{-1}$ which is undefined when $x=0$.

The inverse of the power rule enables all powers of a variable $x$ except $x^{-1}$ to be integrated. This integral is called Cavalieri's quadrature formula and was first found in a geometric form by Bonaventura Cavalieri for $n \ge 0$. It is considered the first general theorem of calculus to be discovered.

$\int\! x^n \,dx= \frac{ x^{n+1}}{n+1} + C, \qquad n \neq -1.$

This is an indefinite integral where $C$ is the arbitrary constant of integration.

The integration of $x^{-1}$ requires a separate rule.

$\int \! x^{-1}\, dx= \ln |x|+C,$

Hence, the derivative of $x^{100}$ is $100 x^{99}$ and the integral of $x^{100}$ is $\frac{1}{101} x^{101} +C$.

## Problems

A simplified method for calculating the power rule formula.

$\frac{d}{dx} x^7 = nx^{7-1} , n=7x^6.$

step 1: multiply 7 x 1 which equals 7 and place the product (answer = 7) in front of "x".

step 2: subtract 7 minus 1 = 6 which equals its exponent.