What is the derivative?

The derivative is used to determine the slope of a function. For a linear function, this is trivial. For example the line defined by has a slope of 2 at any point (or in general, the slope of a line is ). The derivative can be used to find the slope for any point of an arbitrary function. The slope would be the same for the tangent line of a point on the function. If given two points of a function, such as two points of a line, the slope can be calculated as . The goal is to find the slope in a single point. In order to do this, the second point is moved closer and closer to the first one such that the slope calculated using this formula will become more and more precise as the second point approaches the first point. Consider the following image:

In this image, we show a first approximation of the slope at by choosing another point on the function that is distance away form . On the right side of the image, we have a more precise approximation by letting this distance become smaller. If we let approach 0, we will get the exact slope:

The slope is calculated as . This is in accordance with the previously mentioned equation where and and . The derivative of at point is defined as the slope of the tangent of at point and is found by letting approach 0:

For example consider the function . We can calculate the slope or the derivative like this:

(1)

To find the derivative of any function, you need to differentiate it.