Compare two different methods to find a derivative.

1. Use the definition of the derivative as a limit to find the slope function, $f^\prime(x)$, of $f(x) =-x^2 + 3x + 1$.

   Step 1(a):

   Substitute $f(x)$ into the Definition of the Derivative (this is Hana's definition):

   Step 2(a):

   Find a way to 'cancel out the $h$' in the denominator:
   1. Expand the numerator.
   2. Combine like terms.
   3. Factor out an $h$.
   4. Cancel out the $h$.

   Step 3(a):

   Now that there is no longer an $h$ in the denominator, take the limit as $h→0$.
   This is the derivative of $y =-x^2 + 3x + 1$.

2. Use the Power Rule to find $f^\prime(x)$. Do your answers agree?

   Hint (b):

   If your answers are different, then you probably made an algebraic error in part (a). Find it and fix it.

3. Use your slope function to find $f^\prime(0)$ and $f^\prime(1)$.

   Answer (c):

   $f^\prime(0) = 3$, $f^\prime(1) = 1$