Why is there a separate symbol for partial derivatives? The concept of a partial derivative is very simple: for a multivariate function $f$, the partial derivative of $f$ with respect to a single variable $x$ is computed by treating the other variables as constants and differentiating $f$ with respect to $x$.
As a student of Calculus I, I do not fully understand the need for a $\partial y / \partial x$.  As far as I know, the special “partial” $\partial$ does not change the process of the computation.  In addition, I have become confused as the calculus of my physics course increases in difficulty.
Take a three-dimensional position vector $\vec r = \vec x + \vec y + \vec z$ and an electric field vector $\vec E$ that varies with $\vec r$.  If $V$ denotes electric potential, then $\Delta V = -\int \vec E\cdot d\vec r$.
On a review sheet that my teacher created, he expanded this, saying

$$\begin{align}
E_x &= -\partial V / \partial x \\
E_y &= -\partial V / \partial y \\
E_z &= -\partial V / \partial z
\end{align}$$

Given the standard setup of a partial derivative, I see no issue with this.  However, our standard formula chart reads that

$$E_x = -\frac{dV}{dx}$$

and this genuinely confuses me since our calculations of field potential are almost always expanded to multiple dimensions.
What is the need for a $\partial f / \partial x$ notation, and why are sources (at least in physics, the only application of partial derivatives I encounter during the course of the school day) inconsistent?
 A: Alan Turing said:

The Leibniz notation $\mathrm dy/\mathrm dx$ I find extremely difficult to understand in spite of it having been the one I understood best once! It certainly implies that some relation between $x$ and $y$ has been laid down e.g.
  $$ y = x^2 + 3x $$

Whenever the notation $\partial z/\partial x$ is used, there exists some sort of implicit dependency of $z$ w.r.t. $x$, e.g. $z = f(x,y)$. The $\partial$ symbol is used since in this case only the dependency of $z$ w.r.t. $x$ is considered. If there is another implicit dependency $y = g(x)$, $\partial z/\partial x$ does not take that into account (hence the name partial derivative), while $\mathrm dz/\mathrm dx$ does, and it is called the total derivative of $z$ w.r.t. $x$.
A: There is no one forcing you to use $\partial$ for partial derivatives, however it is recommended because it helps avoid confusion in expressions where both partial and ordinary derivatives are present.
A: I would just like to add that I've done a bit more research and that the distinction between $\partial t$ and $dt$ seems vital for formulae like
$$d\left( f\left( x,y,\cdots\right)\right) = \frac{\partial f}{\partial x}dx + \frac{\partial f}{\partial y}dy + \cdots$$
and
$$\begin{align}
\frac{dz}{dx} &= \frac{dz}{dy} \frac{dy}{dx} \qquad \mathrm{\left( chain \ rule\right)} \\
\frac{\partial z}{\partial x} &\neq \frac{\partial z}{\partial y} \frac{\partial y}{\partial x}
\end{align}$$
like in Hurkyl's example
$$\frac{\partial z}{\partial x} \frac{\partial x}{\partial y} \frac{\partial y}{\partial z}
= \left( 5\right) \left( -\frac{3}{5}\right) \left( \frac{1}{3}\right)
= -1$$
if $z = 5x + 3y$
