Home > CALC > Chapter 4 > Lesson 4.3.2 > Problem4-111

4-111.

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 $f(x) =-x^2 + 3x + 1$.

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

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 $h$.

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?

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 $f^\prime(1)$.

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