What do the partial derivates mean in the formulation of total derivate? In the total derivate expression of a function $f(x,y,z)$ where $x, y$ and $z$ are functions of $t$, what do the partial derivates $\frac{\partial f}{\partial x}$,  $\frac{\partial f}{\partial y}$ and  $\frac{\partial f}{\partial z}$ mean? Since $x, y, z$ are functions of $t$, it is impossible to keep one of x, y and z varying and others constant (which is what is done while computing the partially derivatives).
Total derivative of $f$ w.r.t. t:
$$
 \frac{d f}{dt}=\frac{\partial f}{\partial x}\frac{dx}{dt}+\frac{\partial f}{\partial y}\frac{dy}{dt}+\frac{\partial f}{\partial z}\frac{dz}{dt} + \frac{\partial f}{\partial t}
$$
 A: You have to consider different cases, don't confuse them :
FIRST CASE : $\quad f(x,y,z)$
$$df=\frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy+\frac{\partial f}{\partial z}dz$$
SECOND CASE : $\quad f(x(t),y(t),z(t))$
$dx=\frac{dx}{dt}dt \quad;\quad dy=\frac{dy}{dt}dt \quad;\quad dz=\frac{dz}{dt}dt$ .
$df=\frac{\partial f}{\partial x}\frac{dx}{dt}dt+\frac{\partial f}{\partial y}\frac{dy}{dt}dt+\frac{\partial f}{\partial z}\frac{dz}{dt}dt$
$$\frac{df}{dt}=\frac{\partial f}{\partial x}\frac{dx}{dt}+\frac{\partial f}{\partial y}\frac{dy}{dt}+\frac{\partial f}{\partial z}\frac{dz}{dt}$$
THIRD CASE : $\quad f(x,y,z,t)$
$$df=\frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy+\frac{\partial f}{\partial z}dz+\frac{\partial f}{\partial t}dt$$
FOURTH CASE : $\quad f(x(t),y(t),z(t),t)$
$df=\frac{\partial f}{\partial x}\frac{dx}{dt}dt+\frac{\partial f}{\partial y}\frac{dy}{dt}dt+\frac{\partial f}{\partial z}\frac{dz}{dt}dt+\frac{\partial f}{\partial t}dt$
$$\frac{df}{dt}=\frac{\partial f}{\partial x}\frac{dx}{dt}+\frac{\partial f}{\partial y}\frac{dy}{dt}+\frac{\partial f}{\partial z}\frac{dz}{dt}+\frac{\partial f}{\partial t}$$
Don't confuse $\frac{df}{dt}$ and $\frac{\partial f}{\partial t}$
$\frac{\partial f}{\partial t}$ is the partial derivative of $f(x,y,z,t)$ with respect to $t$ with $x,y,z$ not function of $t$.
$\frac{df}{dt}$ is the total derivative of $f(x(t),y(t),z(t),t)$ with $x,y,z$ functions of $t$.
A: Imagine that initially you only had $f$ depending on $x,y,z$ and you had already computed the partial derivatives $\frac{\partial f}{\partial x},\frac{\partial f}{\partial y},\frac{\partial f}{\partial z}$. Suddenly, you realize that in fact $x,y,z$ depend on a single variable $t$. How to compute $\frac{df}{dt}$ without throwing away the derivatives that you already computed? The answer is given by the formula that you mentioned. 
