Why are differential equations used a lot in physics? I have heard from my physics teacher that differential equations are very useful in physics. In what parts of physics exactly is it useful? Why are they generally useful?
 A: I'd like to elaborate on an earlier answer.
In general the quantities that go into the equations come in chuncks, related by differentiation. The most known example is the trio: position, velocity, acceleration. Velocity is the first time derivative of position, acceleration being the second time derivative.
I think the relations we encounter are in fact all of the type where the rate of change of one quantity relates to the magnitude of some other quantity. There is the relation between rate of change of velocity and force: $F=ma$
Then there will be classes of cases where the rate of change of some quantity A relates to rate of change of quantity B.

As we know, equations with one or more derivatives in them are classified as 'differential equations'.
A: Differential equations in physics describe many systems and how they change.  They are everywhere in classical mechanics, they are very common in quantum mechanics and way more. These equations states how a rate of change  in one variable is related to other variables. F=ma is a differential equation, Schrödinger's equation is a differential equation. Many more examples can be made.
The simplest answer to this is that physics is about how fundamental constituents of space interact with each other, and this interaction is explained with differential equations.
A: Spatial gradients drive temporal change.
