3blue1brown touches on that way of thinking about functions/derivatives in this video 

(very similar to Khan academy video linked in the article)
 

In this paragraph:

>In single variable calculus, the chain rule can be written as $\frac{d}{dx}f\left(g\right(x\left)\right)=f^{\prime }\left(g\right(x\left)\right)g^{\prime }\left(x\right)$. In the multivariable case, for a function $f:R^k\to R$, we write $\frac{d}{dx}f\left(g\right(x\left)\right)={\sum }_{i=1}^k\frac{d}{dx}{g}_i\left(x\right)D_{i}f\left(g\right(x\left)\right)$ where $D_i$ is the partial derivative of $f$ with respect to its $i$th argument. This can be simplified by employing the following notation, which uses a dot product: $\frac{d}{dx}f\left(g\right(x\left)\right)=\nabla f\cdot g^{\prime }\left(x\right)$.

What does $g_i\left(\right)$ stand for? 
My understanding that g maps $g:R^m\to R^k$ and $g_i$ indicates i-th element of a vector that g produces.
 

Is that the right way to think about it?