In the book [Pattern Recognition and Machine Learning](https://uwantg.files.wordpress.com/2018/04/bishop-pattern-recognition-and-machine-learning-springer-2006.pdf), the author writes the log-evidence function (equation 3.86 in page 167):

ln $p(\textbf{t}| \alpha, \beta) = \frac{M}{2}$ ln $\alpha$ + $\frac{N}{2}$ ln $\beta$ - $E (\textbf{m}\_N)$ - $\frac{1}{2}$ ln|$\textbf{A}$| - $\frac{N}{2}$ ln($2\pi$)

where $E (\textbf{m}\_N) = \frac{\beta}{2} || \textbf{t} - \Phi \, \textbf{m}\_N ||^2 + \frac{\alpha}{2} || \textbf{m}\_N ||^2 $ (equation 3.82)

Then, he differentiates the log-evidence with respect to $\alpha$ and sets to zero (to maximize), getting:

$0 = \frac{M}{2\alpha} - \frac{1}{2} || \textbf{m}\_N ||^2 - \frac{1}{2} \sum\_i{\frac{1}{\lambda\_i + \alpha}}$ (equation 3.89)

The last term is $\frac{1}{2} \frac{d}{d \alpha}$ ln |$\textbf{A}$|, it is not important in this question and I have already checked it.

The thing is that, as far as I understand, the log-evidence has been differentiated as if $\textbf{m}\_N$ was constant with respect to $\alpha$, which is not the case, because in page 153 we can see its definition:

$\textbf{m}\_N = \beta \;\textbf{S}\_N \; \Phi^T \textbf{t}$ (equation 3.53)

where $\textbf{S}\_N ^{-1} = \alpha \textbf{I} + \beta \; \Phi \Phi^T$ (equation 3.54)

I would appreciate any help to understand the derivation of 3.89 by differentiation of 3.86 with respect to $\alpha$.