How do you differentiate sqrtt*(1-t^2)t(1t2)?

1 Answer
May 19, 2015

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(1t2))=ddt(t12t52)=12t1252t32.

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(1t2))=12t12(1t2)+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}.=12t1212t322t32=12t1252t32.