Derivative Definition

In mathematics, a derivative is a function that measures the rate of change of another function with respect to a variable. In other words, a derivative can be thought of as a tool that tells us how a function is changing at any given point.

There are two ways to think about derivatives: in terms of limits, or in terms of differential calculus. Both approaches ultimately give us the same information, but they take different routes to get there.

The limit approach to derivatives is based on the idea of slope. Slope measures how steep a line is, and we can use it to approximate the derivative of a function at a given point. To do this, we take two points on the graph of the function (one on either side of the point where we want to find the derivative) and calculate the slope between those points. The closer together those points are, the more accurate our approximation will be.

Differential calculus, on the other hand, takes a more direct route to finding derivatives. It relies on the concept of infinitesimals, which are incredibly small numbers that can be used in mathematical calculations just like any other number. Using infinitesimals, we can define derivatives in terms of rates of change rather than slopes.