Power Rule - Differentiation of Arbitrary Polynomials

Differentiation of Arbitrary Polynomials

To differentiate arbitrary polynomials, one can use the linearity property of the differential operator to obtain:

\left( \sum_{r=0}^n a_r x^r \right)' = \sum_{r=0}^n \left(a_r x^r\right)' = \sum_{r=0}^n a_r \left(x^r\right)' = \sum_{r=0}^n ra_rx^{r-1}.

Using the linearity of integration and the power rule for integration, one shows in the same way that

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