Derivative Maxwell boltzmann distribution

1. The problem statement, all variables and given/known data

i need to show that the peak of the maxwell boltzmann distribution is equal to 1/2 kt.



2. Relevant equations

maxwell boltzmann distribution according to modern physics 3rd edition by kenneth kramer.



ill try to do my best with this



[itex] N(E)= \frac{2N}{√∏} \frac{1}{(kT)^\frac{3}{2}} E^\frac{1}{2} e^\frac{-E}{kT}[/itex]



N is the total number of molecules while N(E) is the distribution function (with units energy to the -1) defined so that N(E) dE is the number of molecules dN in the energy interval dE at E. dn=N(E)dE

3. The attempt at a solution



so i need to take the derivative and set that equal to 0 and hope i get 1/2kt. im having trouble with the derivative itself. im taking the derivative with respect to E so everything else is considered a constant. so to try to make this easier i took all that junk in front of the E and said it is just some constant a. that allowed me to go through and do the product rule. after that, ive been trying to simplify it but am getting nowhere. need some advice on how to do this properly as i believe im not.



thanks a bunch