Differentiating an integral wrt a function

OhMyMarkov

Member
Hello everyone!

I've came accross this problem: differentiate $\int _S f \ln f$ with respect to $f$. From previous explanation, I believe $\int _S f \ln f$ means $\int _S f(x) \ln f(x)dx$.

The answer is $\ln f(x)$... Could anyone indicate how they reached this answer?

Thanks!

Klaas van Aarsen

MHB Seeker
Staff member
Hello everyone!

I've came accross this problem: differentiate $\int _S f \ln f$ with respect to $f$. From previous explanation, I believe $\int _S f \ln f$ means $\int _S f(x) \ln f(x)dx$.

The answer is $\ln f(x)$... Could anyone indicate how they reached this answer?

Thanks!
Hi OhMyMarkov! I suspect that should read $\int _S^f \ln f df$.

Suppose the anti-derivative of ln(x) is LN(x), then it follows that:

$\frac{d}{df}(\int _S^f \ln f df) = \frac{d}{df}(\int _S^f \ln x dx) = \frac{d}{df}(LN( f ) - LN( S )) = \ln f$

OhMyMarkov

Member

Okay, now I have the book, please let me give out the exact statement:

$h(f )$ is a concave function over a convex set. We form the functional:

\begin{equation}
\displaystyle J(f )= -\int f\ln f + \lambda _0 \int f + \sum _k \lambda _k \int f r_k
\end{equation}

and "differentiate" with respect to $f(x)$, the $x$th component of $f$, to obtain

\begin{equation}
\displaystyle \frac{\partial J}{\partial f(x)} = -\ln f(x) -1 +\lambda _0 + \sum _k \lambda _k r_k (x)
\end{equation}

Perhaps the problem statement is now clearer...

Klaas van Aarsen

MHB Seeker
Staff member
Hmm, things certainly have changed.

I'm looking at what is some unconventional notation.
Perhaps you can clarify some of it, because I'm guessing a little bit too much.
Your book should define the symbols and notation used somewhere, typically at the beginning of the chapter or the introduction of the book.

From h(f) is a concave function on a convex set, I deduce that f is an element of a convex set.
That suggests that f is not a function, but for instance an element of R^n.
Is it, or could it be a function?

Looking at the results, it appears that $\int f \ln f$ means $\int df \ln f$, that is ln f integrated with respect to f.
Could that be it?
In that case everything appears to work out, except for the "-1"...

For the integrals no boundary is specified.
But the calculation suggests a constant lower bound, perhaps minus infinity, and an upper bound of f, or something like that....?

Can you clarify what f(x), the xth component of f is supposed to mean?