Why is there only one term on the RHS of this chain rule with partial derivatives? I know that if $u=u(s,t)$ and $s=s(x,y)$ and $t=t(x,y)$ then the chain rule is $$\begin{align}\color{blue}{\fbox{$\frac{\partial u}{\partial x}=\frac{\partial u}{\partial s}\times \frac{\partial s}{\partial x}+\frac{\partial u}{\partial t}\times \frac{\partial t}{\partial x}$}}\color{#F80}{\tag{A}}\end{align}$$
A short extract from my book tells me that: 

If $u=(x^2+2y)^2 + 4$ and $p=x^2 + 2y$ then $u=p^2 + 4$ therefore $$\frac{\partial u}{\partial x}=\frac{\partial u}{\partial p}\times \frac{\partial p}{\partial x}\tag{1}$$ as $u=u(x,y)$ and $p=p(x,y)$

The book mentions no origin of equation $(1)$ and unlike $\color{#F80}{\rm{(A)}}$ is has only one term on the RHS; So I would like to know how it was formed. Is $(1)$ simply equivalent to $\color{#F80}{\rm{(A)}}$ but with the last term missing? Or is there more to it than that? 
Many thanks,
BLAZE. 

It comes from the divergence operator $\nabla$.  Let $f$ be a scalar valued function then $\nabla f \equiv \partial \left\langle \dfrac{\partial f}{\partial x}, \dots \right\rangle$ vectorizes $f$.  If you picture $f$ as the height of a hill and its parameters the coordinates of each point of the hill on the Earth, then $\nabla f$ points in the direction of largest change per unit distance on the Earth surface.  So in your example, $s,t$ are the Earth surface coordinates.  Now take the dot product with the tangent vector of a curve on the Earth's surface parameterized by $x: \langle s(x), t(x) \rangle$.  Figure out what that means by looking at what dot product means.  
