Differentials

In a previous lesson we learned that the derivative at a point is defined as the limit, as the change in x approaches zero, of the change in y over the change in x.
Here is a graph of a function f(x). The distance between P and M is Dx, and the distance between M and Q is Dy.
Next let's add a line tangent to the curve at point P. We can define the distance between P and M as the differential dx.

A differential can be thought of as representing a very small variable. Notice that dx is equal to Dx. <--see the link We can define the distance between M and R as the differential dy.
The differential dy is the change in the linearization of the function.
The slope of the tangent line is given by the derivative of the function at point P. Therefore, we can write the differential dy in terms of the derivative f'(x) and the differential dx.
In this equation, dx is an independent variable, and dy is a dependent variable, since dy depends on the values of x and dx.
If we divide both sides of the equation by dx, we obtain an expression of the derivative in terms of the ratio of the differentials dy and dx.

We also can use differentials to estimate the error resulting from approximate measurements.

See link:

http://www.montereyinstitute.org/courses/AP Calculus AB I/course files/multimedia/lesson32/lessonp.html?showTopic=2