84 - The Multivariable Chain Rule

A much more intuitive form of representation of the MV Chain rule is recognizing that the above sum can be interpreted as a dot product, and that's exactly it.
We define the multivariable chain rule in vector form as $$\begin{aligned}\displaystyle
&\frac{d}{dt}f(x(t),y(t)) = \pdv{f}{x}\cdot \dfrac{dx}{dt}+ \pdv{f}{y} \cdot \dfrac{dy}{dt}=\left[\begin{matrix}\displaystyle\pdv{f}{x} \ \displaystyle\pdv{f}{y} \ \vdots\end{matrix}\right]
\cdot
\left[\begin{matrix} \dfrac{dx}{dt} \ \dfrac{dy}{dt} \ \vdots \end{matrix}\right]
\\
&= \nabla f(\vec{v}(t)) \cdot \vec{v}'(t);, \text{which can be expressed in terms of}
\
&\text{the directional derivative as } \nabla_{\large\vec{v}'(t)}f(\vec{v}(t))
\end{aligned}