Differentiation of problems and the Planck function

Please see attached, the first question i think i understand but want to confirm, the second i am getting confused over, please can you help me understand the calulus (product/chain rule) in the differentiation of the Planck function.  

I am not familiar to the Labert W function, and from my text books there is no mention of it, therefore i can only assume that this problem can be differentiated without it.