TITLE: What is the difference between partial and normal derivatives?
QUESTION [35 upvotes]: I have a clarifying question about this question:
What is the difference between $d$ and $\partial$?
I understand the idea that $\frac{d}{dx}$ is the derivative where all variables are assumed to be functions of other variables, while with $\frac{\partial}{\partial x}$ one assumes that $x$ is the only variable and every thing else is a constant (as stated in one of the answers).
Example 1: If $z = xa + x$, then I would guess that
$$
\frac{\partial z}{\partial x} = a + 1
$$
and
$$
\frac{d z}{d x} = a + x\frac{da}{dx} + 1.
$$
since now $a$ should be considered a function.
When we in calculus 1 have $y = ax^2 + bx + c$, then technically we should use $\partial$ as we are assuming $a, b$, and $c$ are constants?
Is this correct?
Example 2: Maybe the thing that is confusing me is that when we do implicit differentiation we use $d$. So if
$$
x^2 + y^2 = 1
$$
then taking $\frac{d}{dx}$ gives
$$
2x + 2y\frac{dy}{dx} = 0
$$
again because $y$ is considered a function.
How would taking $\frac{\partial}{\partial x}$ of an equation like $x^2 + y^2 =1$ work? Does that even make sense?
Example 3: Is it ever possible that using $\partial$ and $d$ can give the same? If, for example $y = x^2$, does it make sense to say that
$$
\frac{\partial}{\partial x} y = 2x?
$$
Edit: My overall question, I guess, is how the notations of partial derivatives vs. ordinary derivatives are formally defined. I am looking for a bit more background.

REPLY [2 votes]: In multivariate calculus when more  than one independent variable $x$ comes into (competing) operation  on  a dependent quantity $z$ , partial derivatives come into definition. When there are two of them in simple cases rate of change  has meaning  when the usually varying second independent quantity $y$ is forced to become constant. That is how partial derivative with respect to first quantity $x$ can be defined. That is, only when $y$  forced to be temporarily constant, can there be a meaning for partial derivatives
$ p=\dfrac{\partial z}{\partial x},q= \dfrac{\partial z}{ \partial y}.$
The definition owes its definition from the Monge's form of surface $ z = f(x,y) $ where slopes $p,q$  are defined for $x$ variation when $y$ is arrested and vice-versa.
For example when differentiating $ \left( \dfrac{z}{x}-y \right)= $ constant,
partially wrt x:  $ \dfrac{x p -z}{x^2}=0 $
partially wrt y:  $ \dfrac{q}{x}-1 =0 $
The partial derivative is always not subservient, it assumes dominant roles eg in physics (electro-magnetics, electro-statics, optics, structural mechanics..) where  they define  a plethora of phenomena through  structured pde to describe propagation in space or material media.
An example from Variational calculus to scale down ( in a non -linear sense) all differential distances in the plane $\sqrt {dx^2 + dy^2}$   or $ \sqrt {r^2 +( r d \theta)^2 } $ by dividing out by a factor $ (r^2 - a^2) $ through a functional. 
When we try to minimize
$$ \int \dfrac{ \sqrt {r^2 +( r d \theta)^2 }}{(r^2 - a^2)}= \int F d\theta $$
we need variation with respect to entire partial  derivative  acting as a full independent variable $\dfrac{\partial r}{ \partial \theta}$  using the Euler-Lagrange Equation: 
$$ F - r^{'}  \dfrac{ \partial F}{\partial {r^{'}}} = C $$ 
$$ \frac{1}{r^2-a^2} \left({ \sqrt {r^2 +( r {'})^2 }} - r^{'}\cdot \frac{r^{'}}{ \sqrt {r^2 +( r {'})^2 }}\right)= \frac{1}{2\lambda}$$
where the constant is adjusted for convenience of later geometric interpretation . After simplification and integration it results in full circles of arbitrary radius $ \lambda$ of eccentric distance $a$ at tangent point.