Deriving Bernoulli's equation via conservation of E So I'm not OK with how some people derive this equation.
These people consider a pipe whose endings have cross-sectional areas and heights which are different. They then use  the conservation of energy principle by saying $dW = dK + dU$  (Where $W$ is work, $K$ is kinetic energy, and $U$ is potential energy).
For this they consider that the work done on the system would be due to external pressure forces exerted on the whole system of water along the pipe. And here comes the part where I disagree: they use this Work to calculate the change in Potential and Kinetic energy for just a small slab of water within the whole system. This is completely invalid isn't it? I mean you would have to consider the entire system, I think.
My way of interpreting the derivation is if you consider just one slab the whole time. Is this a valid way of thinking?
Thanks!
edit: In fact, in one video I saw, the person just says "the middle chunk of water stays the same the whole time, so we can just ignore it".
 A: You have to consider the assumptions that go into deriving Bernoulli's equation in that manner; that the fluid is incompressible, non-viscous, and experiences non-turbulent flow. 
If these are the assumptions under which you've built your model for fluid flow, when you apply some sort of pressure to one end that results in a work on the fluid, because the fluid can't compress at all to absorb the energy nor can it interact with other parts of the fluid to have some resistive drag that causes some of the fluid to behave different then other sections of the fluid, every part of the fluid responds similarly.
Treating one small part of the fluid and its behavior should therefore be the same as any other part of the fluid. Indeed, it's as if the whole fluid is reacting to the force applied to it at once time. This is why Bernoulli's Equation tells us that energy is conserved per unit volume of the fluid, regardless of where it is.
In general, a more rigorous derivation is needed for more complicated fluid models, but that one suffices for the basic dynamics of fluid flow. 
A: You cannot derive the classic Bernoulli Equation from conservation of energy, because, contrary to popular opinion, it is actually not an expression of conservation of energy at all. It is more accurately construed as an integrated expression of the conservation of linear momentum, $F=ma$.
