### Home > CALC > Chapter 3 > Lesson 3.4.3 > Problem3-175

3-175.

Use the definition of a derivative as a limit to find the slope function, $f^\prime(x)$, of $f(x) = 2x^2-3x + 4$. Confirm your slope function with the Power Rule. Then use your slope function to find $f^\prime(3)$ and $f^\prime(−2)$.

Hana's definition of the derivative:
$f'(x) =\lim\limits_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}$

1. Substitute $f(x)$ into Hana's definition
2. Expand the numerator
3. Combine like terms
4. Factor out an $h$
5. Cancel out the $h$
6. Evaluate the limit as $h→0$
This is $f^\prime(x)$

Use the Power Rule to find $f^\prime(x)$. Compare to the $f^\prime(x)$ you found algebraically. If they do not agree, you probably made an algebraic error.

Evaluate $f^\prime(3)$ and $f^\prime-(3)$.