In mathematics, the power rule is one of the most important differentiation rules in calculus. Since differentiation is linear, polynomials can be differentiated using this rule.
The power rule holds for all powers except for the constant value which is covered by the constant rule. The derivative is just rather than which is undefined when .
The inverse of the power rule enables all powers of a variable except to be integrated. This integral is called Cavalieri's quadrature formula and was first found in a geometric form by Bonaventura Cavalieri for . It is considered the first general theorem of calculus to be discovered.
This is an indefinite integral where is the arbitrary constant of integration.
The integration of requires a separate rule.
Hence, the derivative of is and the integral of is .
Read more about Power Rule: Power Rule, Differentiation of Arbitrary Polynomials, Generalizations
Famous quotes containing the words power and/or rule:
“The personal touch between the people and the man to whom they temporarily delegated power of course conduces to a better understanding between them. Moreover, I ought not to omit to mention as a useful result of my journeying that I am to visit a great many expositions and fairs, and that the curiosity to see the President will certainly increase the box receipts and tend to rescue many commendable enterprises from financial disaster.”
—William Howard Taft (1857–1930)
“We’ve only just begun to learn about the water and its secrets, just as we’ve only touched on outer space. We don’t entirely rule out the possibility that there might be some form of life on another planet. Then why not some entirely different form of life in a world we already know is inhabited by millions of living creatures?”
—Harry Essex (b. 1910)