Derivatives and antiderivatives are remarkable ways of getting information about statics or dynamics from the opposite information. For example, physical laws are usually written as relationships between rates of change, but we usually want to know the value of a static variable over time. If we have a function f(x), we denote the derivative function of f(x) by (df/dx)(x). The derivative function in one dimension returns the rate of change of the function in that dimension. The antiderivative functional returns the family of functions that f belongs to.
Ie., Newton's laws, when modernized, tell us that the Force = mass * acceleration, where acceleration is defined as the rate of change of velocity with respect to time, and velocity is defined as the rate of change of position with respect to time. Suppose the force on an object is zero. Then 0 = m*(dv/dt), or dv/dt = 0. The antiderivative of 0 is the family of constant functions, so we know that v(t) = C for some constant C. C may as well be v(0) since v is constant. In other words, no net force implies constant velocity. Similarly, v = dx/dt where x(t) is the 1-dimensional position function. Thus (dx/dt)(t) = v(0). We know that the antiderivative of a constant function f(t)=C is a linear function of the form Ct + D where D is another constant. So we have x(t) = v(0)*t + D. Note x(0) = D, so we have the familiar equation x(t) = v(0)*t + x(0) for the position function of an object with no net force on it. For constant force in 1-dimension, it is easy to get the familiar result from basic physics that x(t) = (1/2)*a(0)*t^2 + v(0)*t + x(0), where a(0) is the constant acceleration. This power of getting a cascade of static relations from a seed of knowledge about rates of change is very important in many sciences. One usually hypothesizes some proportions between derivatives and uses calculus to find out what that entails in empirical terms. The study of solving more difficult relationships between derivatives and their higher-dimensional cousins partial derivatives is the study of differential equations, out of which grew the pure studies of topology and differential geometry.
Using derivatives by themselves gives us information about dynamics when we have information about statics. It is a standard proof in most calculus texts that the derivative of a function (from R into R) at a point is just the slope of the tangent line to the function at that point. More than that, the behavior of the derivative function, being a rate of change over each point, gives us information about the way the function curves. The points at which the derivative function is zero tells us that the function is neither decreasing nor increasing locally, so there is either a peak, a trough, or a temporary landing after which the function will continue its behavior from before. Note that this last condition implies the derivative function of the derivative function should be zero. Thus, looking at the zeros of the second derivative, and the behavior of the second derivative at each of the zeros of the first derivative, allows us to separate out each behavior.
Thus studying the derivative family of a function allows us to say that it has its highest value for some input, lowest value here, and so on. This is also very useful, as beforehand one would have to study reams of algebraic geometry or devise a new method of attack per problem to even start to tackle this task.
In differential geometry, derivatives become a little more abstract, but still correspond to the ordinary derivative representing rate of change when restricted to one dimension. In this case, the derivative becomes the unique linear transformation that vanishes uniformly with the function at each point.