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Limit Definition of the Derivative

We now give a rigorous definition of the derivative, along the lines of the definition of tangent line given above as a limit of certain secant lines.

A secant line for the function f (x) at x = x 0 is a line through the points (x 0, f (x 0)) and (x, f (x)) , for some x in the domain of f . The slope of such a secant line is


The derivative of f at x 0 is the limit of the slopes of the secant lines at x 0 as x approaches x 0 (that is, as the secant lines approach the tangent line). Thus we have the following formula for the derivative of f at x 0 :

f'(x 0) = (x 0) =    

If we let Δx = x - x 0 , the change in x , then x = x 0 + Δx and substitution yields an alternate formula for the derivative:

f’(x 0)) = (x 0) =    

The quotients in the above expressions are often referred to as difference quotients.