PLEASE HELP! I got this problem wrong and I would really like to understand how to do it for future reference. The problem can be found down below. If you would explain step by step, that would be amazing. Thank you so much!

Question

Find the derivative of f(x) = the summation of n = 0 to infinity of (-1)^n x^2n + 1/ 3^2n (n!)^2. Write answer as a summation with lower limit equal to 0.

Answer 1

it's just a power series, so take the derivative of each term.

∞
∑ (-1)^n * 3^(2n)/(n!)^2 x^(2n+1)
n=0

All those factors involving n are just constants,
so the derivative is
∞
∑ (-1)^n * 3^(2n)(2n+1)/(n!)^2 x^(2n)
n=0