The Taylor polynomial of degree 100 for the function \(f\) about \(x=3\) is given by \(P(x)=(x-3)^2 - \dfrac{(x-3)^4}{2!} + \dfrac{(x-3)^6}{3!}+ ... + (-1)^{n+1} \dfrac{(x-3)^{2n}}{n!} + ... - \dfrac{(x-3)^{100}}{50!} \). What is the value of \(f^{(30)}(3)\) ?

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The coefficient for the 30^th derivative can be found with the 15^th term:

$$(-1)^{n+1} \frac{(x-3)^{2n}}{n!} $$

For \(n=15\):

$$ (-1)^{15+1} \frac{(x-3)^{2(15)}}{15!} $$ $$ = \frac{(x-3)^{30}}{15!} $$

Taking the derivative with the power rule 30 times, results in:

$$ = \frac{30\cdot 29\cdot 28...(x-3)^0}{15!} $$ $$ = \boxed{\frac{30!}{15!} } $$