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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 = x0 is a line through the points (x0, f (x0)) 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 x0 is the limit of the slopes of the secant lines at x0 as x approaches x0 (that is, as the secant lines approach the tangent line). Thus we have the following formula for the derivative of f at x0:

f'(x0) = (x0) =    

If we let Δx = x - x0, the change in x, then x = x0 + Δx and substitution yields an alternate formula for the derivative:

f’(x0)) = (x0) =    

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

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