$\begin{array}{cl}
& g(1+h)-g(1) \\
= & \dfrac{1}{\sqrt{5(1+h)+4}}-\dfrac{1}{\sqrt{5(1)+4}} \\
= & \dfrac{1}{\sqrt{5h+9}}-\dfrac{1}{3} \\
= & \dfrac{3-\sqrt{5h+9}}{3\sqrt{5h+9}} \\
= & \dfrac{3-\sqrt{5h+9}}{3\sqrt{5h+9}} \times \dfrac{3+\sqrt{5h+9}}{3+\sqrt{5h+9}} \\
= & \dfrac{3^2-(\sqrt{5h+9})^2}{3\sqrt{5h+9}(3+\sqrt{5h+9})} \\
= & \dfrac{9-5h-9}{3\sqrt{5h+9}(3+\sqrt{5h+9})} \\
= & \dfrac{-5h}{3\sqrt{5h+9}(3+\sqrt{5h+9})}
\end{array}$
Hence, we have
$\begin{array}{cl}
& g'(1) \\
= & \dlim_{h\to 0} \dfrac{g(1+h)-g(1)}{h} \\
= & \dlim_{h \to 0} \dfrac{1}{h} \times \dfrac{-5h}{3\sqrt{5h+9}(3+\sqrt{5h+9})} \\
= & \dlim_{h \to 0} \dfrac{-5}{3\sqrt{5h+9}(3+\sqrt{5h+9})} \\
= & \dfrac{-5}{3\sqrt{5(0)+9}(3+\sqrt{5(0)+9})} \\
= & \dfrac{-5}{3 \times 3 (3+3)} \\
= & \dfrac{-5}{54}
\end{array}$
