Let $g_1 , g_2 : \mathbb{R} \to \mathbb{R}$ be differentiable. Suppose we know that the following derivative exists at some point $x_0$: $$ \frac{d}{dx}[g_1(f_1(x))+g_2(f_2(x))] $$ but do not know if $f_1$ and $f_2$ individually are differentiable at that point.
Can we write the above derivative at that point as follows? $$ \frac{d}{dx}|_{x=x_0}[g_1'(f_1(x_0))\cdot f_1(x) + g_2'(f_2(x_0))\cdot f_2(x)] $$
We alas cannot write the derivative this way. [Updated with more parseable counterexample].
As a counterexample, let $f_1(x)=x^{\frac{1}{3}}$, $g_1(x)=x^3$, and let $f_2\equiv g_2\equiv 0$.
Then let $h(x)=g_1(f_1(x)) + g_2(f_2(x)) = x$, so $h'(0)=1$, yet $$g_1'(f_1(0))\cdot f_1(x)+g_2'(f_2(0))\cdot f_2(x)\equiv 0,$$ and so we have $$\frac{d}{dx}|_{x=0}(g_1'(f_1(0))\cdot f_1(x)+g_2'(f_2(0))\cdot f_2(x))=0.$$