Suppose that an unspecified function f$($x$)$

has the degree $7$

Taylor polynomial at x$=0$

:

p$7($x$)=9$x$716$x$615$x$510$x$4.$

Because

p$7($x$)=$k$=07$f$($k$)(0)$xkk$!$

you can read off the value of the derivatives of f$($x$)$

at x$=0$

by looking at the coefficients of xkk$!$

as follows:

the coefficient of x$11!$

in p$7($x$)$

is f$(1)(0)=$

$0$

the coefficient of x$22!$

in p$7($x$)$

is f$(2)(0)=$

$0$

the coefficient of x$33!$

in p$7($x$)$

is f$(3)(0)=$

$0$

the coefficient of x$44!$

in p$7($x$)$

is f$(4)(0)=$