Existence of the Derivative Function

Example (Existence of Derivative) Give three examples of functions where is not differentiable at but is defined at   

    Solution. The function is not differentiable at since





which proves the two-sided limit (the derivative) does not exist. This type of example where the function is not differentiable is called a corner point.
    Secondly, the function   is not differentiable at since





which proves the two-sided limit (the derivative) does not exist. This type of example where the function is not differentiable is called a vertical tangent.
    Thirdly,  the function is not differentiable at since





which proves the two-sided limit (the derivative) does not exist.

Example (Existence of Derivative)  Compute the difference quotient for the function defined by



Do you think is differentiable at ? If so, what is the equation of the tangnent line at ?

    Solution.  For we find,



At we have, Using a table of values to compute the limit we infer that and so the equation of the tangent line is

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Existence Of The Derivative Function
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/existence-of-the-derivative-function.html