Simplifying [tex]f(x)[/tex] will let you avoid using the quotient rule:
[tex]f(x)=\dfrac{2x^4+1}{7x^4}+3\sqrt[3]{x^2}-4x^{-1/4}[/tex]
[tex]f(x)=\dfrac27+\dfrac17x^{-4}+3x^{2/3}-4x^{-1/4}[/tex]
and now you're just taking derivatives of power functions. You should get
[tex]f'(x)=0-\dfrac47x^{-5}+2x^{-1/3}+x^{-5/4}[/tex]
and we can rewrite this as
[tex]f'(x)=-\dfrac4{7x^5}+\dfrac2{\sqrt[3]{x}}+\dfrac1{x^{5/4}}[/tex]
even combining into one fraction as
[tex]f'(x)=-\dfrac17x^{-5}\left(4-14x^{14/3}-7x^{15/4}\right)=-\dfrac{4-14x^{14/3}-7x^{15/4}}{7x^5}[/tex]
