2022-M2-01

$\begin{array}{cl}& g(1+h)-g(1)\\ =& {\displaystyle \frac{1}{\sqrt{5(1+h)+4}}}-{\displaystyle \frac{1}{\sqrt{5(1)+4}}}\\ =& {\displaystyle \frac{1}{\sqrt{5h+9}}}-{\displaystyle \frac{1}{3}}\\ =& {\displaystyle \frac{3-\sqrt{5h+9}}{3\sqrt{5h+9}}}\\ =& {\displaystyle \frac{3-\sqrt{5h+9}}{3\sqrt{5h+9}}}\times {\displaystyle \frac{3+\sqrt{5h+9}}{3+\sqrt{5h+9}}}\\ =& {\displaystyle \frac{3^2-(\sqrt{5h+9}{)}^2}{3\sqrt{5h+9}(3+\sqrt{5h+9})}}\\ =& {\displaystyle \frac{9-5h-9}{3\sqrt{5h+9}(3+\sqrt{5h+9})}}\\ =& {\displaystyle \frac{-5h}{3\sqrt{5h+9}(3+\sqrt{5h+9})}}\end{array}$

由此，可得

$\begin{array}{cl}& g^{\prime }(1)\\ =& {\displaystyle \underset{h\to 0}{lim}{\displaystyle \frac{g(1+h)-g(1)}{h}}}\\ =& {\displaystyle \underset{h\to 0}{lim}{\displaystyle \frac{1}{h}}\times {\displaystyle \frac{-5h}{3\sqrt{5h+9}(3+\sqrt{5h+9})}}}\\ =& {\displaystyle \underset{h\to 0}{lim}{\displaystyle \frac{-5}{3\sqrt{5h+9}(3+\sqrt{5h+9})}}}\\ =& {\displaystyle \frac{-5}{3\sqrt{5(0)+9}(3+\sqrt{5(0)+9})}}\\ =& {\displaystyle \frac{-5}{3\times 3(3+3)}}\\ =& {\displaystyle \frac{-5}{54}}\end{array}$