Suppose that a function f is defined on some open interval (a,b) on the x-axis.
Then if we choose a fixed point x in this interval and we introduce the difference quotient
f(x+h)-f(x) ————h
where h is a number which may be positive or negative ( but not zero ) and where x+h also lies in the interval (a,b).
The numerator f(x+h) - f(x) measures the change in the function when x changes from x to x+h.
h is allowed to approach zero and if the above quotient approaches some definite value as a limit
(and if this limit is the same whether h approaches zero through positive values or through negative values)
then this limit is called the derivative of f at x
and this is denoted by f '(x) and this is read as "f prime x".