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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.
The integration of requires a separate rule.
Hence, the derivative of is and the integral of is .
Historically the power rule was derived as the inverse of Cavalieri's quadrature formula which gave the area under for any integer . Nowadays the power rule is derived first and integration considered as its inverse.
For integers , the derivative of is that is,
The power rule for integration
for is then an easy consequence. One just needs to take the derivative of this equality and use the power rule and linearity of differentiation on the right-hand side.
Using this, we can see that
Since the division has been eliminated and we have a continuous function, we can freely substitute to find the limit:
The use of the quotient rule allows the extension of this rule for n as a negative integer, and the use of the laws of exponents and the chain rule allows this rule to be extended to all rational values of . For an irrational , a rational approximation is appropriate.
Using the linearity of integration and the power rule for integration, one shows in the same way that
One can prove that the power rule is valid for any exponent r, that is
as long as x is in the domain of the functions on the left and right hand sides and r is nonzero. Using this formula, together with
one can differentiate and integrate linear combinations of powers of x which are not necessarily polynomials.
- Larson, Ron; Hostetler, Robert P.; and Edwards, Bruce H. (2003). Calculus of a Single Variable: Early Transcendental Functions (3rd edition). Houghton Mifflin Company. ISBN 0-618-22307-X.