I don't have good grasp on the Ordinary Differential Equations that were taught last semester, so sorry in advance if this is something I was supposed to learn ahead of time.
So, while going through the topic of genesis of partial differential equations, I came to the following argument, and I have no Idea as to how it came about, here it is:
Let $\phi(x, y, z, a, b) = 0$ be a relation between three variables x,y,z and two arbitrary constants a, b. As usual, z is the dependent variable and x, y two independent variables. In order to require two other equations besides the given relation phi(x, y, z, a, b) = 0 Differentiating the given relation $\phi = 0$ with respect to x we obtain: $$\phi_{x}+\phi_{z}\frac{\partial z}{\partial x}=0$$
I don't understand how the term $\phi_z\dfrac{\partial z}{\partial x}$ comes about.