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