what is the derivative of g(t)=t(4-t)^1/2

g(t)=t(4-t)^1/2        we can use the product and chain rules here

g'(t)  = (4-t)^1/2  + (1/2)t(4-t)^(-1/2)(-1) = (4-t)^1/2 - (1/2)t(4-t)^(-1/2) =

.......(factoring).....   (4-t)^(-1/2) [ (4-t) - (1/2)t]   =  (4-t)^(-1/2)[4- (3/2)t ] =

(4-t)^(-1/2) [(8 -3t)/2]