# Maximum Likelihood problem in book Pattern Recognition and Machine Learning

I ran into a problem in section 2.4.1 namely "Maximum likelihood and sufficient statistics" under "Exponential Distribution Family" of Bishop's Pattern Recognition and Machine Learning. excerpt

How to derive from (2.195) to (2.224)?What am I missing?

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Interchange integration and differentiation. Then just as you would do if η were a scalar d/dη exp(η u(x)) =u(x) exp(η u(x)). –  Michael Chernick Jun 4 '12 at 13:51

The first step is just the product rule for differentiation. That gives 2.224. To get 2.225 move the first term to the other side of the equation and multiple numerator and denominator of the term moved by g(x). Using 2.195 substitute 1 for the lefthand side in 2.195 to get 2.225.

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Ouch..Product rule it is.Then how to differentiate the integral part to get the extra u(x).Or what rule is applied? –  Kooi Nam Ng Jun 4 '12 at 3:51
One more question,how to go from (2.225) to (2.226).Sorry,my math ground was not solid. –  Kooi Nam Ng Jun 4 '12 at 3:58
I figured out the second one but still clueless about the first one. –  Kooi Nam Ng Jun 4 '12 at 4:20
@KooiNamNg see my comment above. –  Michael Chernick Jun 4 '12 at 13:52
@KooiNamNg the last step -del (ln(g(η)) = -del g(η) /g(η). Here del stands for the differential symbol that is used in your text but I don't have. –  Michael Chernick Jun 4 '12 at 14:06