Tide, harsh, and dextercioby are all basically saying the same thing, but let me provide some elucidation. As a historical note, Newton wrote his second law originally in terms of change-of-momentum. That is, the net force acting on an object causes a change in momentum as follows (where the net force and momentum are both vector quantities):
<br />
\vec{F}_{net} = \frac{d}{dt}(\vec{p}) = \frac{d}{dt}(m\vec{v})<br /> whereby \vec{F}_{net} = \sum_i{\vec{F}_i}.
For the non-relativistic case (v << c), we can add the approximation that the mass is a constant and treat it as such. Of course, for relativistic speeds (usually v \ge 0.1c), we need to invoke the special relativity mass correction factor of m = m_0\frac{1}{\sqrt{1-(v/c)^2}}.
How this all applies to differential equations is that the physical phenomenon of nature are modeled by differential equations. Newton's second law above gives a great example. Take the motion of a mass resting on a flat frictionless tabletop that is displaced from its equilibrium position by some deflection x. Now, the primary force we're concerned with is the restoring force of the spring, given by F = -kx where I drop the vector symbols because we only have 1-dimensional freedom of movement and the direction is indicated by sign. Now, by Newton's second law, this gives us the following equation:
<br />
F_{net} = \frac{d}{dt}(mv) \implies -kx = mx'' \implies x'' + \frac{k}{m}x = 0.<br />
Voila. This is a linear second order differential equation with constant coefficients that models the motion of the mass. Two solutions of this differential equation are:
x(t) = \sin{\left(t\sqrt{k/m}\right)} and x(t) = \cos{\left(t\sqrt{k/m}\right)}
and by an elementary theory from differential equations, we know that all solutions of the system can be written in the form of
x(t) = C_1\sin{\left(t\sqrt{k/m}\right)} + C_2\cos{\left(t\sqrt{k/m}\right)}
where C_1 and C_2 are constants determined by the initial conditions of the system. A knowledge of physics and differential equations combined is a powerful tool for producing accurate models of the physical universe.
As for your comment about derivatives and infinities, this is bordering into the philosophy of mathematics, which has a very long history. Debates around infinity and infinitesimals still go on today despite our everyday application of them. Strictly speaking, mathematicians use the term "differential" to describe small changes in one quantity (e.g. dx) whereas they use the term "derivative" to refer to the
operation of differentiation.
Cheers.
---
Mike Fairchild
http://www.mikef.org/
"Euclid alone has looked on beauty bare."
--Edna St. Vincent Mallay
