Question: Consider a conic surface defined by: \begin{equation} z = h \left( 1-\frac{\rho}{a} \right) , \rho = \sqrt{x²+y²} \end{equation} generated by the revolution of the line \begin{equation} z = h \left( 1-\frac{y}{a} \right) , y \geq 0 \end{equation} we want to find the curve with the smallest distance on the conic surface between two points \begin{equation} A(0,-a,0), B(0,a,0) \end{equation}. a) Find the possible functional f, in cylindrical coordinates where: \begin{equation} f(\phi,\phi' \equiv \frac{d\phi}{d\rho},\rho)\end{equation} where $\rho$ is the independent variable on f. Define $\beta$ as $\sqrt{1+\left( \frac{h}{a}\right)^2}$, that must appear on f.
What i did:
Let $ds$ be the distance function in cylindrical coordinates where $(ds)^2 = (dz)^2 + (d\rho)^2 + \rho^2 (d\phi)^2$
We know that: $\frac{dz}{d\rho} = -\frac{h}{a}d\rho$
Then
$(ds)^2 = (dz)^² + (d\rho)^2 + \rho^2 (d\phi)^2 \implies (ds)^2 = \left( -\frac{h}{a}d\rho \right)^2 + (d\rho)^2 + \rho^2 (d\phi)^2 \implies ds = \left( 1 + \frac{h}{a}\right)^2(d\rho)^2 + \rho^2 (d\phi)^2$ Substituting $\beta$ in the equation we have:
\begin{equation*} (ds)² = \beta² (d\rho)² + (\rho)²( d\varphi )² \implies \left(\dfrac{ds}{d\rho}\right)^2 = (\beta)² + (\rho)²\left(\dfrac{d\varphi}{d\rho}\right)^2 \end{equation*}
My question: I read many topics in this subject about why divide or multiply by differential is wrong. And thought that really understood, but i don't know how to apply the right way. I was doing a physics question and the final steps as mentioned that i did was:
\begin{equation*} (ds)² = \beta² (d\rho)² + (\rho)²( d\varphi )² \implies \left(\dfrac{ds}{d\rho}\right)^2 = (\beta)² + (\rho)²\left(\dfrac{d\varphi}{d\rho}\right)^2 \end{equation*}
I would like to write my solution more detailed and mathematically formal. How can i write these steps more detailed and mathematically formal (with rigor of the differentials)?
Edit:Following @PawełCzyż suggestion, i'm adding all the question to get some context.
Cheers