Answer
$y' = \frac{3}{4}x^{-\frac{1}{4}} + 3x^{-4} + \frac{11}{4}x^{\frac{7}{4}}+x^{-2}$
Work Step by Step
Using product rule:
$y = (1+x^{2})(x^{\frac{3}{4}} - x^{-3})$
$y' = (1 + x^{2})(\frac{3}{4}x^{-\frac{1}{4}}+3x^{-4}) + (x^{\frac{3}{4}} - x^{-3})(2x)$
$y' = \frac{3}{4}x^{-\frac{1}{4}} + 3x^{-4} + \frac{11}{4}x^{\frac{7}{4}}+x^{-2}$
Using sum of simpler terms:
$y = x^{\frac{3}{4}} - x^{-3} + x^{\frac{11}{4}} - x^{-1}$
$y' = \frac{3}{4}x^{-\frac{1}{4}} + 3x^{-4} + \frac{11}{4}x^{\frac{7}{4}}+x^{-2}$