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Differentiation of Problems and the Planck Function

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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 calculus (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.

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This solution assists in solving the differentiation problems.

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