Question

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

Answer 1

It's quickest to multiply `\sqrt{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}.`

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)`

`=\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}.`