A function f of one real variable is said to be differentiable at argument x, if its graph looks like a straight line for arguments in any open interval including x. (An open interval is one that does not contain its endpoints.)
Its derivative at x is the slope of that line.
(To be more precise, for whatever positive criterion of nearness you choose however small, there is an open interval containing x so that for every x' in the interval other than x itself, the difference between and the slope of that line is less than that criterion.) Note
The line that f resembles near argument x is called the tangent line to f at argument x and the linear function it represents is called the linear approximation to f at argument x.
The slope of the tangent line at x is given by on that line for any two point P1 and P2 on it, with P1 = (P1x, P1y) and P2 = (P2x, P2y)
df = P2y- P1y
dx = P2x - P1x
We use the notation dx and df to denote changes in the corresponding variables that are so small that we can assume the linear approximation to f (and to any other function involved in the definition of f) is exacty satisfied (and if there is no such distance create one in your imagination).
Changes of this sort are called differentials. The derivative of f at argument x is usually written as
In fact if df and dx are differentials, then the derivative is by definition since the derivative is the ratio of the change of f to the change of x in the linear approximation.
The applet here allows you to enter any standard function and domain, and look at it, its slope and derivative in it.