Sorry if my language is not precise. Let $f_i:\mathbb{R}^m\to \mathbb{R}^n$ be some differentiable functions in the sense of auto-diff. How can you construct new functions $g(f_1,f_2,f_3,...)$ such that g is also differentiable? And what is the branch of mathematics concerned with this kind of question? some obvious examples I'm aware of are: Additions: $g=f_1+f_2+...$, polynomials:$g=a_1f_1+a_2f_1^2+a_3f_1^3+b_1f_2+b_2f_2^2+b_3f_2^3+...$, compositions:$g=f_1(f_2(f_3))$, I'm particularly interested functions involving integrals of $f$ such as $(\int{f}-C)^2$ where C is some constant. How do I learn to construct these new differentiable functions from first principles? Thanks!
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