shouldn't the n=1 be n=2 at the highlighted part? as it was differentiated twice?
Normally, when we differentiate a power series, we lose a term off the front because it's constant. But in this case, the powers are all even. Since there is no $x^1$ term in $f(x)$, there is no constant term in $f'(x)$. We're counting by $2$'s here, so only every other differentiation erases a term.