Finding a functional derivative

Question

I'm a bit stuck on finding the derivative of a functional $H[f]=\int g[f(x)]dx$. I've started off with $$\frac{\delta H[f]}{\delta f(x_0)}=\lim_{\epsilon\to 0} \frac{1}{\epsilon}\bigg[\int g[f(x)+\epsilon\delta (x-x_0)]dx-\int g[f(x)] dx\bigg]$$ I think that I should probably put it into some form similar to $$\frac{\delta H[f]}{\delta f(x_0)}=\lim_{\epsilon\to 0} \frac{1}{\epsilon}\bigg[\int g[f(x)]+[\epsilon\delta (x-x_0)]*\mathrm{something}\ dx-\int g[f(x)] dx\bigg]$$ so that the $\int g[f(x)]dx$ parts cancel out leaving it as $$\frac{\delta H[f]}{\delta f(x_0)}=\int \delta (x-x_0)]*\mathrm{something}\ dx$$however I have no idea how to get whatever "something" should be. (Possibly some part of functions I've forgotten), thanks.

Answer 1

I'm going to use a mathematically better definition than the physicist definition you are using: $$ \left< \frac{\delta H[f]}{\delta f(x)}, \phi(x) \right> = \left. \frac{d}{d\lambda} H[f+\lambda\phi] \right|_{\lambda=0} $$ where $\phi$ is a "test function".

By the chain rule we have $$\begin{align} \frac{d}{d\lambda} H[f+\lambda\phi] & = \frac{d}{d\lambda} \int g[f(x)+\lambda\phi(x)] \, dx \\ & = \int \frac{d}{d\lambda} g[f(x)+\lambda\phi(x)] \, dx \\ & = \int g'[f(x)+\lambda\phi(x)] \, \phi(x) \, dx \\ \end{align}$$ so $$ \left< \frac{\delta H[f]}{\delta f(x)}, \phi(x) \right> = \left. \int g'[f(x)+\lambda\phi(x)] \, \phi(x) \, dx \right|_{\lambda=0} = \int g'[f(x)] \, \phi(x) \, dx = \left< g'[f(x)], \phi(x) \right> $$

Thus, $$ \frac{\delta H[f]}{\delta f(x)} = g'[f(x)] $$