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)?