We covered gradients, partial derivatives, directional derivatives and several basic properties:
$f$ is differentiable at $x$ and $\nabla f(x) = m$ iff:
$$ \lim_{h \to 0}\frac{ f( x+ h) - f( x) - m \cdot h}{\|h\|} = 0. $$
If $f$ is differentiable at $x$ then all directional derivatives exist at $x$ and:
$$ \partial_{ u}f( x) = u \cdot \nabla f( x). $$
In particular, the gradient (if exists) is the vector of partial derivatives:
\left(\frac{\partial f}{\partial x_1}, \ldots, \frac{\partial f}{\partial x_d}\right)( x) . $$
Existence of partial derivatives does not guarantee differentiability, but if partial derivatives are continuous ($C^1$ class) then $f$ is differentiable
Geometrically, $\nabla f(x)$ is the direction in $\R^d$ of steepest increase of $f$; and at a (local) minimum/maximum, $\nabla f(x)=0$
Chain Rule deals with differentiation of composition of functions.
Consider single variable functions $f,g$, and define their composition $h(x) = f(g(x))$. The familiar form of the Chain Rule says that
$$ h'(x) = f'(g(x)) \cdot g'(x). $$
Another way to write this is as follows: define a real variable $y$ related to $x$ via $y = g(x)$, and then
$$ \frac{dh}{dx} = \frac{df}{dy} \cdot \frac{dy}{dx}. $$
(Here $dy/dx = dg/dx = g'(x)$.)
This admits a very powerful generalization to multi-variable functions…