Derivative

The Derivative of a Function at a point is a measure of the rate at which that function is changing as one of the Variables changes. This corresponds to the slope of the graph of the function at that point.

Intuitively, suppose we wish to find the derivate of a suitably smooth Function, f(x) say, at the point x=a. If we increase x by a small amount, which we'll call dx (or more accurately delta x) we can calculate f(a + dx). An approximation to the slope is given my (f(a+dx)-f(a)) / dx, which is to say it is the change in f divided by the change in x. The smaller the value dx, the better approximation.

Mathematically, we define the derivative, denoted f'(a), to be the Limit of this ratio as dx tends to zero.