Differentiation is defined in many ways. Geometrically, it is finding the slope of a line drawn tangent to a function. This involves taking smaller and smaller increments between points on a graph so that the final value approaches the same number (the limit of a function). Physically, it is calculating the instantaneous velocity or acceleration of an object in motion. Put in more simple language, a derivative measures the rate of change of quantity A as quantity B changes.
The clearest thing about the history of differentiation is that there is no clear "inventor" of the subject. Rather it was a culmination of lifetimes of work. Early Greek and Far Eastern mathematicians made great use of "infinitesimals," very small measurements originally used in finding volumes and areas. In the late 1600s, Isaac Newton and Wilhelm Gottfried Leibniz independently discovered the relationship between the derivative and the integral, which was then formulated into the Fundamental Theorem of Calculus. But they alone did not unravel differentiation. Newton and Leibniz built upon work by Isaac Barrow, Christiaan Huygens and René Descartes, among others.
As you learn the fundamentals of differentiation, you are introduced to different methods of differentiating. In cases where it is impossible or tedious to solve a derivative using one variable, implicit differentiation is used. This is the process of solving the derivative using a non-standard variable. For example, instead of solving for y in an equation, you solve for x. Partial differentiation, introduced usually in a second semester calculus course, is used when you need to solve for one variable while defining another variable as a constant. Finally, higher order differentiation is often used in physics classes to determine both the velocity and acceleration of a moving object. It involves taking the derivative of the same function more than once.
Differentiation has a number of real-world applications. In business, derivatives are used to find the maximum profit potential or the minimum production cost of an item. Also using the minimum/maximum idea is optimization, where city planners, builders and even farmers can determine the best use of their resources and space. Related rates are used by scientists, communications experts and economists to track the movement of different variables. For example, economists can track the marginal revenue of a new product or the population growth of an entire group of people.