How is $dx \over dy$ different from $\partial x \over \partial y$? Say I have variables $x,y_1,y_2,z_1,z_2$ all $\in \mathbb{R}$
And I have the following equations:
$$x = f_1(y_1,y_2)$$
$$y_1 = f_2(z_1,z_2)$$
How does:
$$dx \over dz_1$$
differ from:
$$\partial x \over \partial z_1$$
or am I confused?
Intuitively I just want to think about how $x$ varies in proportion to an infinitesimally small perturbation of $z_1$, so I don't understand the difference between the two different notations (nonpartial vs partial)?
 A: When a function is defined on more than one variables we use $\cfrac{ \partial f }{\partial x}$ to denote the partial derivation of $f$ with respect to one of its variables $x$ while holding the other variables constant.
This is a question of notation. Using $\cfrac {df} {dx}$, the total derivative, will show anyone that sees it that $f$ is only defined on the variable $x$ or that the other variables of $f$ are functions also defined on $x$.


*

*Say for example a function $f(x_1, x_2, x_3)$ such that $x_1$, $x_2$, $x_3$ are independent. Then $$\cfrac {\partial f}{\partial x_1} =\left ( \cfrac {\partial f}{\partial x_1}\right )_{x_2, x_3}$$ is the partial derivative of $f$ with respect to $x_1$ holding $x_2, \space x_3$ constant.

*But consider a function $f(t,x_1(t), x_2(t), x_3(t))$ then
$$ \cfrac {df}{dt} = \cfrac {\partial f}{\partial t}\cfrac {dt}{dt}+\cfrac {\partial f}{\partial x_1}\cfrac {dx_1}{dt}+\cfrac {\partial f}{{\partial x_2}}\cfrac {dx_2}{dt}+\cfrac {\partial f}{{\partial x_3}}\cfrac {dx_3}{dt}$$ is the total derivative of $f$ with respect to $t$.

*Now let's compute the total derivative of the first function $f(x_1, x_2, x_3)$ with respect to $x_1$
$$\cfrac {df}{dx_1} =\cfrac {\partial f}{\partial x_1}\cfrac {dx_1}{dx_1}+\cfrac {\partial f}{\partial x_2}\cfrac {dx_2}{dx_1}+\cfrac {\partial f}{{\partial x_3}}\cfrac {dx_3}{dx_1} = \cfrac {\partial f}{\partial x_1}$$ which is basically the same thing but only because the variables are independent of one another. It changes if one of them say $x_3$ is a function of $x_1$ then $\cfrac {dx_3}{dx_1} \ne 0$ and then $$\cfrac {df}{dx_1} = \cfrac {\partial f}{\partial x_1} + \cfrac {\partial f}{{\partial x_1}}\cfrac {dx_3}{dx_1} $$
See Partial Derivatives and Total Derivatives. 
