Geometrically, the derivative of a function f (x) at a point x = c represents the slope of the tangent to the curve y = f (x) at (c,f (c)).

Let y = f (x) be differentiable on an open interval containing c. Let p (c,f (c) ) and Q (c+h) be two neighboring points on the graph of y = f (c). Then the slope of the secant line PQ is f (c+ h) - f(c) Since
                                                                                                                          h
the tangent at P is a limiting position of secant lines PQ, the slope of the tangent at P is the limiting value of the slope of the secant lines as Q approaches P. But as Q approaches p, h → 0.


The slope of the tangent at P

= lim  f (c+ h) - f(c)
             h

= f ’(c).

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