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)?