Leibniz's Derivative Notation

Differentiation is a method to compute the rate at which a dependent output y changes with respect to the change in the independent input x. This rate of change is called the derivative of y with respect to x.

In terms of a graph the derivative is represented by the slope of the line tangent to the curve at a point.

From introductory Algebra you know that the slope of a line through two points is defined as the "change in y divided by the change in x", or in symbols Δy/Δx.

Putting these ideas together, the slope of a line requires two points, yet a derivative is the slope of the tangent line-- only one point! Problem!

The solution to this problem is to think of letting the change in x become as small as we like, in other words we take the limit of Δy/Δx as ∆x goes to 0. As we do this the slope of the line passing through points A and B gets closer and closer to the slope of the tangent.

An elegant notation that captures the essence of this process was created by Leibniz. He used dy/dx to represent Δy/Δx AFTER the limit has been taken. In other words, he used dy/dx as the notation for the derivative of the function.

Move point B closer to point A to get an illustration of this idea and the notation.

Trivial, but pleasing, point: Leibniz changes the "D" from the Greek version to the Latin version to represent taking a limit; this happens again when the Greek "S" (sigma of Riemann sum notation) becomes the elongated Latin "S" of the integral symbol.

Jeff Holcomb (with help from M. Poirier), Created with GeoGebra