Marginal Costs production with labor only

Hello everyone, the question might sound a bit too trivial but I am really stuck on this in my modeling attempt and need a second opinion on it. Is the following derivation of marginal costs correct?

1-Gamma is just a function affecting the output but is not related to labor and A is labor productivity.

My production function has the following form:

Total real costs of the production are

And the resulting marginal costs in real terms

Thanks a lot in advance!

Yes, it looks correct.