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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 df = P2f - P1f , 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
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