Search Menu

Contents

Limit Definition of the Derivative

Limit Definition of the Derivative

Limit Definition of the Derivative

Limit Definition of the Derivative

Limit Definition of 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

   

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.