If f and g are infinitely differentiable functions , can I garentee that fog is infinitely differentiable functions?(I am not pretty sure whether it is true however). I am pretty sure about the case of kf,f+g,fg(Leibniz rule).
I wonder a proof, if it is sufficiently easy. I don't have so much mathematics background, but reading the book Elementary Analysis The theory of Calculus (Kenneth A. Ross). If the proof is too difficult, maybe I would like to ask which field of mathematics does it mostly related to, and what should I learn/ which book(s) should I read in order to understand the proof.
Thanks.
By the Product and Chain Rules, $$\frac{d}{dx}f(g(x))=f'(g(x))\cdot g'(x)$$ $$\frac{d^2}{dx^2}f(g(x))=f''(g(x))\cdot g'(x)^2+f'(g(x))\cdot g''(x)$$ $$\frac{d^3}{dx^3}f(g(x))=f'''(g(x))\cdot g'(x)^3+3f''(g(x))\cdot g'(x)\cdot g''(x)+f'(g(x))\cdot g'''(x)$$ et cetera so the answer is yes.