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About

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 Derivatives07:05

The Power Rule for Derivatives

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