What is exactly difference between $(\partial f)/\partial x$ and $df/ dx$ , Where f=f(x,y,z)? I have a problem about meaning partial derivative.Question is following;
What are the meaning difference between $(\partial f)/\partial x$ and $df/ dx$ , Where $f=f(x, y, z)$ ?
For example , $f$ is area of a triangle respect to $x$, $y$, $z$  and we want to find rate of change area respect to $z$. Should we use $(\partial f)/\partial z$ or $df/ dz$ ?
 A: Both notations are used to denote the derivative, although the one with $\partial$ is more common (and the one with a roman $d$ is almost always reserved either for a function of a single variable, or for situations like
$$
f(s, t) = s^2 + t^3\\
z = f(x, x^2)
$$
Now one speaks of $\frac{dz}{dx}$, and sometimes folks will simply write something like
$$
f(x, y) = x^2 + y^3\\
z = f(x, x^2)
$$
and then informally call $z$ by the name $f$ and write $df/dx$ (alas). I find this completely baffling, but friends who work in Physics do this all the time. 
Sadly, the fact is that these things are ambiguous. For instance, if
$$
f(x, y, z) = x + \sin z
$$
and 
$$
g(x, y, z) = f(y, x, z)
$$
then the application of the chain rule to compute 
$$
\frac{\partial g}{\partial x}
$$
would say that it's 
$$
\frac{\partial f}{\partial x}\frac{\partial z}{\partial x} + \frac{\partial f}{\partial y}\frac{\partial x}{\partial x} + \frac{\partial f}{\partial z}\frac{\partial z}{\partial x}
$$
where the first two partials of $f$ are almost meaningless. There's a lot to be said for instead using
$$
D_1 f
$$
to denote the derivative of $f$ with respect to its first argument, so that 
if 
$$
f(x, y, z) = x + \sin z \\
g(z, y, x) = z + \sin x
$$
then (since $g$ and $f$ are the same function) we have
$$
D_1 f (3, 2) = D_1 g(3, 2)
$$
while in the Leibniz notation, we have
$$
\frac{\partial f}{\partial x}= \frac{\partial g}{\partial z}
$$
and vice-versa. 
The downside is that the chain rule in the "partials" form is so very easy to remember. 
A: The notation depends on the number of variables of a function. If the function has only one variable then you write 
$\frac{dy}{dx}$
for the first derivative w.r.t. $x$.
If the function has more than one variable then you write 
$\frac{\partial y}{\partial x}$
for the first (partial) derivative w.r.t. $x$.
