I am calculating the maximum differential entropy by Lagrange multiplier.
And I found this in wiki but I cannot understand why there is 1 between $ln(g(x)), \lambda_{0}$, I cannot figure out why it comes out after comparing the two equations (I know that the constants are all discarded).
Thank you.
The derivative of $x\mapsto x\ln x$ is $x\mapsto 1+\ln x$. So when you compute the functional derivative of $F[g]=\int_{-\infty}^{\infty}g(x)\ln(g(x))\mathrm dx$ with respect to $g$, you compute $$\delta F[g]=\int_{-\infty}^{\infty}\left[\left(g(x)+\delta g(x)\right)\ln(g(x)+\delta g(x))-g(x)\ln(g(x))\right]\mathrm dx=\int_{-\infty}^{\infty}\left[\ln(g(x))+1\right]\delta g(x)\mathrm dx+o(\delta g^2).$$ Therefore the functional derivative is $$\frac{\delta F}{\delta g(x)}=\ln g(x)+1.$$