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.
