It's quickest to multiply \sqrt{t}√t through with the distributive property and then use linearity and the power rule:
d/dt(\sqrt{t}\cdot (1-t^{2}))=d/dt(t^{1/2}-t^{5/2})=\frac{1}{2}t^{-1/2}-\frac{5}{2}t^{3/2}.ddt(√t⋅(1−t2))=ddt(t12−t52)=12t−12−52t32.
For extra fun, you can also use the product rule along with linearity and the power rule:
d/dt(\sqrt{t}\cdot (1-t^{2}))=\frac{1}{2}t^{-1/2}(1-t^{2})+t^{1/2}\cdot (-2t)ddt(√t⋅(1−t2))=12t−12(1−t2)+t12⋅(−2t)
=\frac{1}{2}t^{-1/2}-\frac{1}{2}t^{3/2}-2t^{3/2}=\frac{1}{2}t^{-1/2}-\frac{5}{2}t^{3/2}.=12t−12−12t32−2t32=12t−12−52t32.