In calculus [Calculus], the power rule is used to differentiate [Differentiate] functions of the form \(f(x) = x^r\), whenever \(r\) 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 \(f\) be a function satisfying \(\displaystyle f(x)=x^{r}\) for all \(x\), where \(r\in {\mathbb {R}}\). Then,

\(\displaystyle f’(x)=rx^{r-1}\)

The power rule for integration states that

\(\int\! x^r \, dx=\frac{x^{r+1}}{r+1}+C\)

for any real number \(r \neq -1\). It can be derived by inverting the power rule for differentiation. In this equation \(C\) is any constant.

(“Power Rule” 2022)