I've been working on my Math IA for a while, its a project part of the IB course which requires me to write a ~20 page math paper investigating and exploring something within maths. I chose to write my paper on using the Euler Lagrange equation to solve for the function which minimizes the surface area of a bubble between two rings, aka a catenoid.
I did all the math for the paper a while ago, and I'm revisiting it to write the introduction and conclusion. While rereading the math, something didn't make sense to me. My final answer was correct, but a step didn't make sense or just didn't seem right.
A section of my working out (which I took from a couple sources) starts off with this equation, from which we have to solve for $f$.
$$f'\sqrt{1+(f')^2} - f'\frac{d}{dx} \left( \frac{ff'}{\sqrt{1+(f')^2}} \right)=0$$
and from here we can replace the first $f'$ with $\frac{d}{dx}f$, giving us:
$$\frac{d}{dx}f\sqrt{1+(f')^2} - f'\frac{d}{dx} \left( \frac{ff'}{\sqrt{1+(f')^2}} \right)=0$$
and then for some reason, I factorise the operator $\frac{d}{dx}$ to get:
$$\frac{d}{dx} \left[ f\sqrt{1+(f')^2} - \left( \frac{f(f')^2}{\sqrt{1+(f')^2}} \right) \right]=0$$
and from there I just integrate, solve for $f'$, and solve the separable diff equation to solve for $f$ in terms of $x$ ($\cosh$).
The step I am confused about is when I factorize out the $\frac{d}{dx}$, since its an operator. The math works because I got the correct answer, but why does it work, to me that shouldn't work but for some reason it does.
Is there a reason for why this works? If so what is it.
We have in play a function $G(u)=\sqrt{1+u^2}$ with $G'(u)=\frac{u}{G(u)}$. Your initial expression becomes $$ f'G(f')-f'[fG'(f')]' $$ With the first step you intend to assemble the full derivative $$ [fG(f')]'=f'G(f')+ff''G'(f') $$ while the second term is part of the full derivative $$ [ff'G'(f')]'=f'[fG'(f')]'+ff''G'(f'). $$ As you can see, the additional term is the same in both cases, so it cancels in the difference and you get indeed that the original expression is a full derivative $$ [fG(g')-ff'G'(f')]'. $$