Use the definition of the derivative to write an expression (i.e. the limit of a difference quotient) representing each of the following derivatives. Then evaluate each limit to find the actual derivative. L'Hôpital's Rule may be used.

1. $f^\prime(x)$, where $f ( x ) = \frac { 1 } { 2 x + 1 }$.

   Hint 1(a):

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

   Steps (a):

   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 $h$ in the denominator and then evaluate as $h→0$.
   2) Deconstruct Hana's Method. Once you identify $f(x)$, find $f^\prime(x)$ 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 $0$ in the numerator and the denominator.

   Hint 2(a):

   

2. $h^\prime(x)$, where $h(x) = (x − 2)^{−2}$.

   Hint (b):

   Refer to the hints in part (a).

3. $g^\prime(4)$, where $g ( x ) = \sqrt { x }$.

   Hint (c):

   Refer to the hints in part (a).