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
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“Power Rule.” 2022. Wikipedia, November. https://en.wikipedia.org/w/index.php?title=Power_rule&oldid=1119651193.