What is the operator "capital D" and how can the chain rule be used in this way I ought to know this but I was somehow always able to avoid mixing partials and $d$s 

My book notes the following: (with some intermediate steps which are less important than the time taken to write)

Here $f$ is a function of $x(t),y(t),z(t),t$ and $u=\frac{dx}{dt}$ and "" for others.

$\frac{Df}{Dt}=\frac{\partial f}{\partial t}+(\mathbf{u}\cdot\nabla)f$

I was taught to use $D$ for total derivative but there's no malformed interpretation I could apply here. 

What is going on and if you were to look it up in an index of a book what would the book be on, and what would you look up? (E.G: Real analysis, partial differentiation)

Also I'm not sure how to use bold with LaTeX nor get a special dot. If someone could tell me in the comments I'd be very grateful.
This is just the chain rule.  It may help to think of this as follows:

We have
$f = f(x(t),y(t),z(t),w(t))$ where it so happens that $w(t) = t$.  We then have
$$
\newcommand{\pwrt}[2]{\frac{\partial #1}{\partial #2}}
\newcommand{\dwrt}[2]{\frac{d #1}{d #2}}
D_tf = \pwrt fw \dwrt wt  +
\pwrt fx \dwrt xt + 
\pwrt fy  \dwrt yt + 
\pwrt fz  \dwrt zt =\\
\pwrt fw + (\mathbf u \cdot \nabla)f
$$
you may consider rewriting $\pwrt fw$ as $\pwrt ft$ a slight abuse of notation.
It's just the chain rule.

To be more precise, write $x=X(t)$, $y=Y(t)$, $z=Z(t)$ in order to distinguish the quantities $x$, $y$, $z$ from the functions $X$, $Y$, $Z$ which say how they depend on $t$. Now take your function $f(x,y,z,t)$ and define
$$
g(t) = f(X(t),Y(t),Z(t),t)
.
$$
Then the notation $Df/Dt$ (by definition) means exactly $g'(t)$, which can be computed using the chain rule:
$$
\begin{align}
g'(t) &amp;= f'_x(X(t),Y(t),Z(t),t) \, X'(t) \\ &amp;+ f'_y(X(t),Y(t),Z(t),t) \, Y'(t) \\ &amp;+ f'_z(X(t),Y(t),Z(t),t) \, Z'(t) \\ &amp;+ f'_t(X(t),Y(t),Z(t),t),
\end{align}
$$
which equals the given expression since $X'(t)=u$, etc.