# How do you find the derivative of a power series?

Oct 2, 2014

One of the most useful properties of power series is that we can take the derivative term by term. If the power series is

$f \left(x\right) = {\sum}_{n = 0}^{\infty} {c}_{n} {x}^{n}$,

then by applying Power Rule to each term,

$f ' \left(x\right) = {\sum}_{n = 0}^{\infty} {c}_{n} n {x}^{n - 1} = {\sum}_{n = 1}^{\infty} n {c}_{n} {x}^{n - 1}$.

(Note: When $n = 0$, the term is zero.)

I hope that this was helpful.