The following formulas for $f_{-}^{\prime }\left(a\right)$ and $f_{+}^{\prime }\left(a\right)$ represent the left- and right-sided derivatives of a function at a point $a$, respectively:
$f_{-}^{\prime }\left(a\right)=\underset{h\to 0^{-}}{\lim }\frac{f(a+h)-f\left(a\right)}{h}$, $f_{+}^{\prime }\left(a\right)=\underset{h\to 0^{+}}{\lim }\frac{f(a+h)-f\left(a\right)}{h}$
Consider $f\left(x\right)=\{\begin{array}{c}6-x^2\text{ if }x\le 2\\ 3x-4\text{ if }x>2\end{array}$. Find $f_{-}^{\prime }\left(a\right)$ and $f_{+}^{\prime }\left(a\right)$ at $a=2$.

• A. $f_{-}^{\prime }\left(2\right)=-4$$f_{+}^{\prime }\left(2\right)=3$
• B. $f_{-}^{\prime }\left(2\right)=-4$$f_{+}^{\prime }\left(2\right)=-3$
• C. $f_{-}^{\prime }\left(2\right)=6$$f_{+}^{\prime }\left(2\right)=3$
• D. $f_{-}^{\prime }\left(2\right)=6$$f_{+}^{\prime }\left(2\right)=-3$

Answer: (A)