Áp dụng công thức tính đạo hàm , ta có:

a) $y'=\left [ (2x^{2}-x+1)^{\frac{1}{3}} \right ]'$

= $\frac{1}{3}(2x^{2}-x+1)^{(\frac{1}{3}-1)}(2x^{2}-x+1)'$

= $\frac{4x-1}{3.(2x^{2}-x+1)^{\frac{2}{3}}}$

b) $y'=\left [ (4-x-x^{2})^{\frac{1}{4}} \right ]'$

= $\frac{1}{4}(4-x-x^{2})^{(\frac{1}{4}-1)}(4-x-x^{2})'$

= $\frac{-2x-1}{4.(4-x-x^{2})^{\frac{3}{4}}}$

c) $y'=\left [ (3x+1)^{\frac{\prod}{2}} \right ]'$

= $\frac{\prod }{2}(3x+1)^{(\frac{\prod }{2}-1)}(3x+1)'$

= $\frac{3\prod }{2}(3x+1)^{(\frac{\prod }{2}-1)}$

d) $y'=\left [(5-x)^{\sqrt{3}}  \right ]'$

= $\sqrt{3}(5-x)^{(\sqrt{3}-1)}(5-x)'$

= $-\sqrt{3}(5-x)^{(\sqrt{3}-1)}$