Q: Rearranging gradients in the chain rule I wish to find the following 
\begin{align*}
\frac{\partial f(x)}{\partial g(x)}
\end{align*}
where $x\in\mathbb{R}^n$, $f,g:\mathbb{R}^n\to\mathbb{R}$. Using the chain rule, with $y=g(x)$, as
\begin{align*}
\frac{\partial f(x)}{\partial x} = \frac{\partial f(x)}{\partial y}\frac{\partial y}{\partial x}
\end{align*}
So I was thinking of writing $\frac{\partial f(x)}{\partial y} = \left(\frac{\partial f(x)}{\partial x}\right)/\left(\frac{\partial y}{\partial x}\right)$, but I don't think this is right because both of the terms on the right hand side are vectors. How do I rearrange terms when the terms are multi-dimensional?
