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Use the definition of derivative as a limit to write an expression representing each of the following derivatives. Then evaluate each expression to write the actual derivative. l’Hôpital’s Rule may be used.

  1. , where 

    Setup the definition of the derivative as a limit.
    This is Hana's Method:

    Make a choice about how to evaluate this limit. These are your choices:
    1) Use algebra. Simplify the fraction until you can 'cancel out' the in the denominator and then evaluate as .
    2) Deconstruct Hana's Method. Once you identify , find using the Power Rule. Of course, in this case, that is working in circles.
    3) Use l'Hôpital's Rule; after all, there is a in the numerator and the denominator.

  2. , where

    Refer to the hints in part (a).

  3. , where

    Refer to the hints in part (a).