### Home > APCALC > Chapter 4 > Lesson 4.3.2 > Problem4-122

4-122.

Compare two different methods for determining a derivative in parts (a) and (b).

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

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

$f'(x)=\lim \limits_{h\rightarrow 0}\frac{(-(x+h)^{2}+3(x+h)+1)-(-x^{2}+3x+1)}{h}$

Find a way to 'cancel out the $h^\prime$ 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 write an equation for $f^\prime$. 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 calculate $f^\prime\left(0\right)$ and $f^\prime(1)$.

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