We are given to solve for t to maximise W,

W=(1-(1/(t+1)))^a*(1/(t+1))^b*e^-tE

where t>0 or t=0

Which can be log linearised to obtain

lnW=aLn(1-(1/(t+1)))+bLn(1/(t+1))-tE

Obviously to obtain the maximimum we need to differentiate the above with respect to t and find the zero value(s) for the differential, (and then locate which is the maximum) but I am stuck on how to proceed with the differentiation.

Any help would be most appreciated.