Concept# Power rule

Summary

In calculus, the power rule is used to 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.
Statement of the power rule
Let f be a function satisfying f(x)=x^r for all x, where r \in \mathbb{R}. Then,
: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.
Proofs
Proof for real exponents
To start, we should choose a working definition of the value of f(x) = x^r, where r

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