So I'm trying to understand why we can use expansions of Maclaurin series in this form.
If I try to convert the following into a Maclaurin Series $$f(x) = x^3 \cos(x^2)$$ Using the following: $$\cos(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!}$$ I get $$x^3 \cos(x^2) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{4n+3}}{(2n)!}$$
Why can we plug in $(x^2)$ in in the place of $\cos(x)$? Isn't there a chain rule to consider, since the Maclaurin/Taylor Series have something to do with derivatives? Why is it that we can just plug in $x^2$ and it still works?