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

1 Answer
Vinicius M. G. Silveira
Dec 16, 2014

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).

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