\(h\left(x\right)=\dfrac{1}{2}\cos\left(\dfrac{2\sqrt{x}+4}{\sqrt{x}-3}\right)+\dfrac{1}{2}\)
\(\Rightarrow h'\left(x\right)=\dfrac{1}{2}\left[-\sin\left(\dfrac{2\sqrt{x}+4}{\sqrt{x}-3}\right)\right].\left(\dfrac{2\sqrt{x}+4}{\sqrt{x}-3}\right)'=-\dfrac{1}{2}.\dfrac{\left(2\sqrt{x}+4\right)'\left(\sqrt{x}-3\right)-\left(2\sqrt{x}+4\right)\left(\sqrt{x}-3\right)'}{\left(\sqrt{x}-3\right)^2}.\sin\left(\dfrac{2\sqrt{x}+4}{\sqrt{x}-3}\right)\)
\(=-\dfrac{1}{2}.\dfrac{\dfrac{1}{\sqrt{x}}\left(\sqrt{x}-3\right)-\dfrac{1}{2\sqrt{x}}\left(2\sqrt{x}+4\right)}{\left(\sqrt{x}-3\right)^2}\sin\left(\dfrac{2\sqrt{x}+4}{\sqrt{x}-3}\right)=-\dfrac{1}{2}.\dfrac{-\dfrac{3}{\sqrt{x}}-\dfrac{2}{\sqrt{x}}}{\left(\sqrt{x}-3\right)^2}\sin\left(\dfrac{2\sqrt{x}+4}{\sqrt{x}-3}\right)\)
