Question

How do you use a Maclaurin series to find the derivative of a function?

Answer 1

The MacLaurin series of a function `f` is a power series of the form:

`sum_(n=0)^(oo) a_n x^n`

With the coefficients `a_n` given by the relation

`a_n=(f^((n))(0))/(n!),`

where `f^((n))(0)` is the `n`th derivative of `f(x)` evaluated at `x=0`.

Therefore,

`f^((n))(0)=a_n n!`

This reasoning can be extended to Taylor series around `x_0`, of the form:

`sum_(n=0)^(oo) c_n (x-x_0)^n`

With the relation

`f^((n))(x_0)=c_n n!`

It's important to emphasize that the function `n`th derivative of `f` (that is, `f^((n)) (x)`) cannot be obtained directly from the Taylor/MacLaurin series (only it's value on the point around wich the series is constructed).