Maybe it would help to think of it in terms of two coordinate systems:
1. The "laboratory" coordinate system, x,y,z
2. A moving coordinate system x',y',z' that moves (and deforms) with the flowing fluid particles
We are going to do a transformation between the two coordinate systems to see what we get.
We're going to let the laboratory coordinate x represent the position at time t of a fluid particle that was at position x=x', y=y', z=z' at time t = 0. Similarly for y and z. So were are going to express the laboratory coordinates of all the fluid particles at time t parametrically in terms of their laboratory coordinates at time t = 0. In that way, the coordinates of the particles at time zero act as labels (identifiers) for the material particles which they carry along with them for all times. So, the trajectory of each particle of fluid can be represented as:
x = x(t,x',y',z')
y= y(t, x',y',z')
z= z(t, x',y',z')
with
x' = x(0,x',y',z')
y'= y(0, x',y',z')
z'= z(0, x',y',z')
This is simply a transformation between a moving coordinate system and a fixed coordinate system. The kinematics of the fluid flow are totally specified once the functions x(t,x',y',z'), y(t,x',y',z'), and y(t,x',y',z') are specified.
Using this framework, let's start out by determining the velocities of all the fluid particles at time t. To do this, we first write,
dx=\frac{\partial x}{\partial t}dt+\frac{\partial x}{\partial x'}dx'+\frac{\partial x}{\partial y'}dy'+\frac{\partial x}{\partial z'}dz'
Now, if we are going to determine the velocity of a fluid particle, we are going to have to be holding its "material coordinates" x', y', and z' constant. Therefore, we must have that
v_x=\left(\frac{\partial x}{\partial t}\right)_{x',y',z'}
Similarly for the fluid velocities in the y and z directions.
Next, let's suppose that the temperature or the concentration of the flowing fluid is varying with time and position in the laboratory reference frame. In the case of concentration, for example, we can write C = C(t,x,y,z) and
dC=\frac{\partial C}{\partial t}dt+\frac{\partial C}{\partial x}dx+\frac{\partial C}{\partial y}dy+\frac{\partial C}{\partial z}dz
so that, at a given location in space (x, y,and are constant), the rate of change of concentration with respect to time is given by:
\frac{\partial C}{\partial t}=\left(\frac{\partial C}{\partial t}\right)_{x,y,z}
Next, let's determine what happens if we examine the rate of change of concentration with respect to time as measured by an observer traveling along with a material particle (i.e., holding x', y', and z' constant):
\left(\frac{\partial C}{\partial t}\right)_{x',y',z'}=\left(\frac{\partial C}{\partial t}\right)_{x,y,z}+\left(\frac{\partial C}{\partial x}\right)_{x,y,z}\left(\frac{\partial x}{\partial t}\right)_{x',y',z'}+\left(\frac{\partial C}{\partial y}\right)_{x,y,z}\left(\frac{\partial y}{\partial t}\right)_{x',y',z'}+\left(\frac{\partial C}{\partial z}\right)_{x,y,z}\left(\frac{\partial z}{\partial t}\right)_{x',y',z'}
But, from our equation for velocity, we then have:
\left(\frac{\partial C}{\partial t}\right)_{x',y',z'}=\left(\frac{\partial C}{\partial t}\right)_{x,y,z}+v_x\left(\frac{\partial C}{\partial x}\right)_{x',y',z'}+v_y\left(\frac{\partial C}{\partial y}\right)_{x',y',z'}+v_z\left(\frac{\partial C}{\partial z}\right)_{x',y',z'}
This is just the definition of the material derivative of C with respect to time.
\frac{DC}{Dt}=\left(\frac{\partial C}{\partial t}\right)_{x',y',z'}=\left(\frac{\partial C}{\partial t}\right)_{x,y,z}+v_x\left(\frac{\partial C}{\partial x}\right)_{x',y',z'}+v_y\left(\frac{\partial C}{\partial y}\right)_{x',y',z'}+v_z\left(\frac{\partial C}{\partial z}\right)_{x',y',z'}
