Maclaurin Series of cos(x)

Question

Using the Maclaurin Series of cos(x) ${\displaystyle \cos x=\sum_{n=0}^\infty(-1)^n\frac{x^{2n}}{(2n)!}}\quad\textrm{for }|x|<\infty.$ Consider $\displaystyle f(x)=x^2\cos\big(\dfrac{x^2}{2} \big)$ find the value of $f^{(10)}(0)$. My guess was to expand out the series to obtain the new Maclaurin series form of f(x) and then find the coefficient of the 10th term. I was able to expand out the series to obtain $$f(x) = \sum\limits_{n=0}^{\infty}(-1)^n\frac{x^{2+4n}}{2^{2n}(2n)!}$$ but am unsure of how to proceed further.