# Chain rule with functional derivative

## Main Question or Discussion Point

Given that

$$F = \int{f[h(s),s]ds}$$

does

$$\frac{\partial}{\partial h}ln(F)=\frac{1}{F}\frac{\delta F}{\delta h}=\frac{1}{F}\frac{\partial f}{\partial h}$$

?

## Answers and Replies

Ben Niehoff
Gold Member
Given that

$$F = \int{f[h(s),s]ds}$$

does

$$\frac{\partial}{\partial h}ln(F)=\frac{1}{F}\frac{\delta F}{\delta h}=\frac{1}{F}\frac{\partial f}{\partial h}$$

?
I don't think that means anything now. Shouldn't you be calculating

$$\frac{\delta}{\delta h(s')} \textrm{ln}(F)$$

with some fixed $$s'$$?

I don't think that means anything now. Shouldn't you be calculating

$$\frac{\delta}{\delta h(s')} \textrm{ln}(F)$$

with some fixed $$s'$$?

If so, then is

$$\frac{\delta}{\delta h(s')} \textrm{ln}(F) = \frac{1}{F[h(s')]}\frac{\delta F}{\delta h(s')}$$

?

Basically I'm not sure if the same kind of chain rule applies to functional derivatives as to regular ones. I haven't been able to find an example of a derivative of a function containing a functional like this natural log one.

If so, then is

$$\frac{\delta}{\delta h(s')} \textrm{ln}(F) = \frac{1}{F[h(s')]}\frac{\delta F}{\delta h(s')}$$

?
$$\frac{\delta}{\delta h(s')} \textrm{ln}(F[h]) = \frac{1}{F[h]}\frac{\delta F[h]}{\delta h(s')}$$

F depends only on the mapping h, not on variable s. It seems to be okey to leave the parameter $$h$$ (which is a mapping itself) out, and write $$F=F[h]$$, but $$F[h(s')]$$ would not make sense.

It's like here. If you have $$f:\mathbb{R}^3\to\mathbb{R}$$, and $$x\in\mathbb{R}^3$$, then you can write $$f(x)$$, but $$f(x_3)$$ (from $$x=(x_1,x_2,x_3)$$) would not make sense.

Basically I'm not sure if the same kind of chain rule applies to functional derivatives as to regular ones. I haven't been able to find an example of a derivative of a function containing a functional like this natural log one.
I have the same problem. I have never encountered a good source on functional differentiation. I can merely calculate, which is often enough, though. If you keep your head clear about simple things like what are parameters for each function, there are not many alternatives left for calculation rules.

Thanks

Ben Niehoff
$$\frac{df}{dx} = \lim_{\Delta x \rightarrow 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}$$