I am having trouble taking the derivative of the following:

t(4-t^2)^(1/2)

I can't decide if you need to use the product rule or not.

What I did is used the PR to get

(4-t)^1/2)*1t^2*(4t-1)^(-1/2)

Is this correct?

you do.

f= t(4-t^2)^.5
f'= (4-t^2)*.5 + .5t/(4-t^2)^.5 * 2t/1

I can't reconcile yours with mine

if g=uv
g'= u'*v+ v'*u