Answer
(a) $\frac{1}{2\sqrt {a+6}}$
(b) $\frac{1}{4\sqrt {2}}$, $\frac{1}{4}$
Work Step by Step
(a) $$f'(a)=\lim_{h\to0}\frac{f(a+h)-f(a)}{h}=\lim_{h\to0}\frac{\sqrt {a+h+6}-\sqrt {a+6}}{h}=\lim_{h\to0}\frac{(\sqrt {a+h+6}-\sqrt {a+6})(\sqrt {a+h+6}+\sqrt {a+6})}{h(\sqrt {a+h+6}+\sqrt {a+6})}=\lim_{h\to0}\frac{a+h+6-(a+6)}{h(\sqrt {a+h+6}+\sqrt {a+6})}=\lim_{h\to0}\frac{1}{\sqrt {a+h+6}+\sqrt {a+6}}=\frac{1}{2\sqrt {a+6}}$$
(b) $f'(2)=\frac{1}{2\sqrt {2+6}}=\frac{1}{4\sqrt {2}}$ and $f'(-2)=\frac{1}{2\sqrt {-2+6}}=\frac{1}{4}$
