What is a derivative?

To put it simply, derivatives are slopes. 

A derivative can also be written as f '(x). 

If x here is a particular point, such as f ' (3), then the result will be a slope of the function when it is at x =3

If it is only f ' (x), then this represents the derivative of the whole function. Ie, f ' (x) is a function that can be used to find any slope on the original function, and from that, the exact point on the original function can be found. 

You can also call f ' (x) the instantaneous slope of points. 

If we pick two points on the function f (x), point x and point x+h, is it possible to find the slope between these two points.

m = [ f (x+h) - f (x) ] / ( x + h - x )

    = [ f (x+h) - f (x) ] / h

Because the slope m is the derivative f ' (x), we can write....

This is also another definition for a "derivative".

[to be continued]