$\partial_i (dx^j/ds)=0?$

Question

Consider the curve $x^i=x^i(s)$, where $i=1,2,...,N$, $x^i$ are coordinates, not power indices.

Is $$\frac{\partial}{\partial x^i}\left(\frac{dx^j}{ds}\right)=0$$?

It wouldn't make any sense to use the chain rule to convert $s$ back to $x^i$, no?

Answer 1

For a counterexample let $s$ be $\theta$ on the circle $x^2+y^2=1$ so$$x=\cos s,\,y=\sin s\implies\frac{\partial}{\partial x}\frac{dy}{ds}=\frac{\partial}{\partial x}x=1.$$