Help with calculation with derivatives.

Question

Ok, so this is probably a brain fart on my part, but anyway I have that $x=e^s$, and the second step of the following is unclear to me: \begin{equation} \frac{d}{ds}=\frac{d}{dx}\frac{dx}{ds}=x\frac{d}{dx} \end{equation} My question is why is $\frac{dx}{ds}=x$?

Answer 1

Because: $$\frac{{\rm d}\color{red}{x}}{{\rm d}s} = \frac{\rm d}{{\rm d}s}(\color{red}{e^s}) = e^s = x.$$