Suppose $f:\mathbb{R^n}\to \mathbb{R^m}$ is differentiable at $x^*$ and $g:\mathbb{R^m}\to \mathbb{R^k}$ is differentiable at $f(x^*)$.
I am trying to prove $g \circ f$ is differentiable at $x^*$ with derivative $(g\circ f)'(x^*)=g'(f(x^*))f'(x^*)$.
The hint of this problem is to use the definition of differentiability for $f,g,$ and $g\circ f$ to derive an error term for $(g\circ f)'$ in terms of those for $f$ and $g$.
Definition : A function $f:\mathbb{R^n}\to \mathbb{R^m}$ is differentiable t $x^*$ if there exists an m$\times$n matrix $f'(x^*)$ and a function $r_{x^*}:\mathbb{R^n}\to \mathbb{R^m}$ such that all $x\in \mathbb{R^n}$. we have $f(x)=f(x^*)+f'(x^*)(x-x^*)+\|x-x^*\|r_{x^*}(x)$, where $\lim_{x\to x^*} r_{x^*}(x)=0$. In other words, $f$ can be approximated by the linearlization $x\to f(x^*)+f'(x^*)(x-x^*)$.
What I have been trying to do is use the idea of proving single variable Chain rule, because I thought it would be similar. But I am having a hard time to find ... Am I on the right track? What type of error term should I look for?
I would really appreciate your help.