I have trouble finding the derivative of the differential entropy w.r.t the probability density function, i.e. what is  $\frac{\delta F[p]}{\delta p(x)}$, where $F[p] = \int_X p(x)\ln(p(x))dx$, and $p(x)$ is a probability density function. 

I tried finding the derivative with the definition of the Frechet derivative, but was not able to solve it. Also, there already exist many posts on the derivative of the related Shannon Entropy, but I was not able to find anything on the differential entropy. 

Any ideas?

Thanks,
Jan