Power rule

In calculus, the power rule is used to differentiate functions of the form , whenever is a real number. Since differentiation is a linear operation on the space of differentiable functions, polynomials can also be differentiated using this rule. The power rule underlies the Taylor series as it relates a power series with a function's derivatives. Let be a function satisfying for all , where . Then, The power rule for integration states that for any real number . It can be derived by inverting the power rule for differentiation. In this equation C is any constant. To start, we should choose a working definition of the value of , where is any real number. 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. It is therefore preferable to use a functional definition, which is usually taken to be for all values of , where is the natural exponential function and is Euler's number. First, we may demonstrate that the derivative of is . If , then , where is the natural logarithm function, the inverse function of the exponential function, as demonstrated by Euler. Since the latter two functions are equal for all values of , their derivatives are also equal, whenever either derivative exists, so we have, by the chain rule, or , as was required. Therefore, applying the chain rule to , we see that which simplifies to . When , we may use the same definition with , where we now have . This necessarily leads to the same result. Note that because does not have a conventional definition when is not a rational number, irrational power functions are not well defined for negative bases. In addition, as rational powers of −1 with even denominators (in lowest terms) are not real numbers, these expressions are only real valued for rational powers with odd denominators (in lowest terms).