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:
“One cannot demand of a scholar that he show himself a scholar everywhere in society, but the whole tenor of his behavior must none the less betray the thinker, he must always be instructive, his way of judging a thing must even in the smallest matters be such that people can see what it will amount to when, quietly and self-collected, he puts this power to scholarly use.”
—G.C. (Georg Christoph)
“There are two great rules in life, the one general and the other particular. The first is that every one can in the end get what he wants if he only tries. This is the general rule. The particular rule is that every individual is more or less of an exception to the general rule.”
—Samuel Butler (1835–1902)