10.89. y = ( (x^3 - 1)^(1/5) ) * arctg x

Use the product rule: $$y' = ((x^3 - 1)^{1/5})' arctg x + (x^3 - 1)^{1/5} (arctg x)'$$

Calculate derivatives: $$((x^3 - 1)^{1/5})' = \frac{1}{5}(x^3 - 1)^{-4/5} (3x^2)$$ and $$(arctg x)' = \frac{1}{1+x^2}$$.

Substitute and simplify: $$y' = \frac{3x^2}{5(x^3 - 1)^{4/5}} arctg x + \frac{(x^3 - 1)^{1/5}}{1+x^2}$$.