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.

Power rule

If is a function such that , and is differentiable at , then,

The power rule for integration, which states that

for any real number , may be derived by applying the Fundamental Theorem of Calculus to the power rule for differentiation.

Proof

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 .