Are these two derivatives the same with regards to TR? Given that Marginal Revenue is change in Total Revenue: 

TR = P*Q 
P = f(Q)
TR = f(Q) * Q
MR = dTR/ dQ


Or 

TR = P*Q 
Q = f(P)
TR = f(P) * P
MR = dTR/dP


Which of these approach is right? My end goal is to relate revenue to elasticity. Does it matter how I arrive to it?
Both of these approaches are correct and valid approaches, in the first approach you calculate the derivative of total revenue with respect to quantity in second with respect to price.

If you are interested in seeing how total revenue varies with quantity you can use the first one if you want to see how the total revenue varies with price then the second one.

Also, you can have multiple different elasticities. Elasticity for single variable function is just by definition $EL=\frac{f’(x)x}{f(x)}$. Depending on what types of elasticity you want to relate the total revenue and what  exactly is your ultimate goal of your study either of the formulas could be used.