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Choose from more than 900 textbooks from leading academic publishing partners along with additional resources, tools, and content. Centennial Campus, 1791 Varsity Dr. Get the latest tips, news, and developments. Please forward this error screen to 68. Although it is feasible to define the value as the limit of a sequence of rational powers that approach the irrational power whenever we encounter such a power, or as the least upper bound of a set of rational powers less than the given power, this type of definition is not amenable to differentiation. This necessarily leads to the same result.

For the two cases that do exist, the values agree with the value of the existing power rule at 0, so no exception need be made. 0 as y approaches 0. Thus, it would be problematic to ascribe any particular value to it, as the value would contradict one of the two cases, dependent on the application. It is traditionally left undefined. Let n be a positive integer. Therefore the base case holds.

Suppose the statement holds for some positive integer k, i. By the principle of mathematical induction, the statement is true for all positive integers n. At the time, they were treatises on determining the area between the graph of a rational power function and the horizontal axis. With hindsight, however, it is considered the first general theorem of calculus to be discovered. 17th century, who both then used it to derive the power rule for integrals as the inverse operation.