My question concerns a map \begin{align} \gamma:A\subset\mathbb R\to\mathbb R^2,\qquad t\mapsto \gamma(t)=(x(t),y(t)) \end{align} such that $x:A\to\mathbb R$ and $y:A\to\mathbb R$ are differentiable at an interior point $t_0\in A$ and such that $(\text{d} x/\text{d} t)(t_0)\neq 0$. Let us call this situation (1).
I was wondering what are the minimal additional conditions that $x,y$ must satisfy in order to be able to conclude that the real-valued function $f$ defined by $x(t) \mapsto y(t)$ is well-defined in a neighborhood of $x(t_0)$ and differentiable at $x(t_0)$ with \begin{align} \frac{\text{d} f}{\text{d} x}(x(t_0)) = \frac{(\text{d} y/\text{d} t)(t_0)}{(\text{d} x/\text{d} t)(t_0)}\quad . \end{align}
Let us call this result (2). I managed to prove that the following condition is sufficient, but I'm wondering if there is a less restrictive sufficient condition.
Sufficient condition 1. If, in situation (1), $x$ is continuous and injective in a neighborhood of $t_0$, then result (2) holds.
In particular, this implies another sufficient condition.
Sufficient condition 2. If, in situation (1), $x$ is $C^1$ in a neighborhood of $t_0$, then result (2) holds.
If there is a less restrictive (but still simple) sufficient condition, I would love to hear about it.