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 equation has a slope of 2 at any point (or in general, the slope of a line equation is equation). 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 equation can be calculated as equation. 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:

derivative1

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

derivative2

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

    equation

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

(1)   equation

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