Differentiating an integral wrt a function

[OhMyMarkov]

Hello everyone!

I've came across 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!

 

[I like Serena]

Hello everyone!

I've came across 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]

Hello ILikeSerena, thanks for replying!

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...

 

[I like Serena]

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?

 

[blue_raver22]

Hello there!

Differentiating an integral with respect to a function is known as the Leibniz integral rule. In this case, we have an integral of the form $\int_S f(x) \ln f(x) dx$, where $S$ is some region of integration. To differentiate this with respect to $f(x)$, we can use the Leibniz integral rule, which states that:

$\frac{d}{df(x)} \int_S f(x) g(x) dx = \int_S \frac{\partial}{\partial f(x)} (f(x)g(x)) dx = \int_S g(x) dx$

Applying this rule to our integral, we get:

$\frac{d}{df(x)} \int_S f(x) \ln f(x) dx = \int_S \frac{\partial}{\partial f(x)} (f(x)\ln f(x)) dx = \int_S \ln f(x) dx$

Therefore, the answer is indeed $\ln f(x)$. I hope this helps! Let me know if you have any further questions.