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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.

References


Links

Video

The Power Rule for Derivatives

The Power Rule for Derivatives