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Let y = f (x) be a function defined in an interval (a,b). i.e. x = c be any point  of this interval so that f (c) is the corresponding value of the function. Let c + h be any other point of this interval which lies to the right or left of c according as h is positive or n-gative. The corresponding value of the function is f (c+h). Then f (c+h) – f (c) is the change in the dependent variable y corresponding to the change h in the independent variable x. Consider the ratio

f (c+ h) - f (c)

Of these two changes which is a function of h and is not defined for h = 0, c being a fixed point.

Definition.  A function y = f (x) is said to be differentiable at x = c if

lim f (c + h) - f (c)
h→0    h

exists and the limit is called the derivative of the function f (x) at x = c. This limit is denoted by the symbol f’ (c). Other notations for f’ (c) are (dv/dx)   and y1 (c).
                                                                  x = c

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