Hier ist meine Lösung:

\( f(x)=(\frac{4x^{-\frac{1}{2}}+\frac{1}{4}x^\frac{3}{2}}{x^2})^{-\frac{3}{2}}
= (4x^{-\frac{5}{2}}+\frac{1}{4}x^{-\frac{1}{2}})^{-\frac{3}{2}}
= (\frac{4}{x^{\frac{5}{2}}}+\frac{1}{4x^{\frac{1}{2}}})^{-\frac{3}{2}}\\
= (\frac{16}{4x^{\frac{5}{2}}}+\frac{x^2}{4x^{\frac{5}{2}}})^{-\frac{3}{2}}
= (\frac{16+x^2}{4x^{\frac{5}{2}}})^{-\frac{3}{2}}
= (\frac{4x^{\frac{5}{2}}}{16+x^2})^{\frac{3}{2}}
= \frac{8x^{\frac{15}{4}}}{(16+x^2)^{\frac{3}{2}}}\\
f'(x)=\frac{6 x^\frac{11}{4} (80 + x^2)}{(16 + x^2)^\frac{5}{2}} \)