taylor expansions for composition with a non differentiable function

Question

Suppose we have a taylor series expansion for f(x), and g(x) is continuous in the interval of convergence of f but it is not differentiable at some points. Can we use the taylor expansion for f(g(x)) by replacing x by g(x) in the taylor expansion of f(x) at those points where g is not differentiable? But then considering f(g(x)) as a new function we won't be able to obtain the derivatives...eg. f(g(x))=cos(|x|) . (In this case the error function also tends to zero). Thomas calculus early transcendentals book just states that g(x) needs to be continuous over the interval, but i wanted to get some clarity over this