Differentiation and Derivatives

Differentiation is the process used to find a derivative. It involves finding a function whose value at every point is equal to the slope of some other function . By performing one or more of a few simple operations on the initial function, we can acquire this second function — the derivative function. We can then use this second function to determine the value of a derivative at any point along the graph provided the initial function is smooth and continuous. So what are derivatives? Why do we wish to calculate them?

So what are derivatives anyway?

In the image below we have a function p(t) representing the position of some object — say, a car, with position on the vertical axis and time along the horizontal axis. At time t = 0 the car’s position is at p = 0. As we approach time t=1 the car’s position has changed slightly to about p = 0.2. At time t=2 the position is changing more drastically, indicating an increasing velocity, until we arrive at the midpoint of the upward sloping curve. Here our velocity is at its maximum and the slope of the graph is steepest. Finally, at about t=4 the car comes to rest at position p = 2 and stays there indefinitely.