I realize that the link would probably be too complex for you to grasp at this time which I why I mentioned that the derivation is quite complex and lengthy. Bernoulli's Equation is then derived from the full equations by eliminating terms which we choose to assume are negligible (read: engineers way of saying ignore).
Even if you don't wish to understand the derivation of the full equations or want to sit down and do it yourself, it's still important to understand them so you can see where the derived forms come from.
They look hard, but essentially they are the fundamental laws of physics, put into fluid perspective: Conservation of Mass, Conservation of Momentum, and Conservation of Energy. Assuming no viscous effects let's you ignore LOTS of terms, which gives you the Euler Eqeuations.
One can then further reduce down terms, maybe the problem is 1D, letting you ignore any \frac{\partial}{\partial y} and \frac{\partial}{\partial z} terms, etc etc.
As far as
partial differentiation, typically it's used as a starting point for a numerical analysis (at least for me). When numerically solving fluid flows, we look at the equations and see them in the form of:
\frac{\partial \vec{Q}}{\partial t} + \frac{\partial \vec{E}}{\partial x} + \frac{\partial \vec{F}}{\partial y} + \frac{\partial \vec{G}}{\partial z} = S
We then say that if we can just find spatial derivatives of the flux vectors E,F,G, then we can "march" in time by computing the time derivative to obtain the new "Vector of Conserved Variables", the Q vector which is mass,momentum and energy. Starting with some initial condition and proceeding in time until the solution converges gives us our answer.
As far as analytic solutions, I learned them in a class called Conduction. It seems like it would be Heat Transfer II, but rather it's a class about solving PDE's, due to the fact that they appear so much in heat transfer. Solving these equations analytically are quite difficult and require a significant amount of prerequisite math before attempting IMHO.
