[tex]f(x)=x^3-\frac{1}{3x^5}+2\sqrt{x}-\frac{3}{x}+\frac{1}{x^3}-\frac{2x}{x^3}\\\\f(x)=x^3-\frac{x^{-5}}{3}+2x^{\frac{1}{2}}-3x^{-1}+x^{-3}-2x^{-2}\\\\f'(x)=3x^2+\frac{5}{3}x^{-6}+x^{\frac{-1}{2}}+3x^{-2}-3x^{-4}+4x^{-3}\\\\f'(x)=3x^2+\frac{5}{3x^6}+\frac{1}{\sqrt{x}}+\frac{3}{x^2}-\frac{3}{x^4}+\frac{4}{x^3}[/tex]
[tex]y=(x+\frac{1}{x})^2-\frac{5}{\sqrt{3x}}\\\\y=(x+x^{-1})^2-5(3x)^{\frac{-1}{2}}\\\\y'=2(x+x^{-1}).(x+x^{-1})'+\frac{5}{2}(3x)^{\frac{-3}{2}}(3x)'\\\\y'=2(x+x^{-1})(1-x^{-2})+\frac{5}{2}(3x)^{\frac{-3}{2}}.3\\\\y'=2(x+x^{-1})(1-x^{-2})+\frac{15}{2}(3x)^{\frac{-3}{2}}\\\\y'=2(x-x^{-1}+x^{-1}-x^{-3})+\frac{15}{2}(3x)^{\frac{-3}{2}}\\\\y'=2(x-x^{-3})+\frac{15}{2}(3x)^{\frac{-3}{2}}\\\\y'=2(x-\frac{1}{x^3})+\frac{15}{2\sqrt{(3x)^3}}\\\\y'=2x-\frac{2}{x^3}+\frac{15}{6x\sqrt{3x}}\\\\[/tex]
[tex]y'=2x-\frac{2}{x^3}+\frac{5}{2x\sqrt{3x}}[/tex]
