I'm reading Harbovsky and Susskind's The Theoretical Minimum Vol I.

In Lecture 3: Dynamics, under Aristotle's Law of Motion is mentioned Aristotle's (fallacious) law $F(t) = m {dx\over dt}$. Later, time is reversed by "changing $t$ to $-t$" and the result is mentioned as $F(-t) = -m{dx\over dt}$. This made absolutely no sense to me. So I tried to formulate this in precise mathematical terms. The following is my result.

Let $\tau : \mathbb{R\to R}$ be a function defined by $\tau (t):=-t$ and let $\dot{f}$ denote the derivative of a differentiable function $f$. Let $F$ and $x$ be differentiable functions satisfying $F(t) = m\dot x(t)$ for some constant $m$. Then $F(-t) =-m\dot{(x\circ\tau)}(t)$.

Question: Am I correct?

I'm reading Harbovsky and Susskind's The Theoretical Minimum Vol I.

In Lecture 3: Dynamics, under Aristotle's Law of Motion is mentioned Aristotle's (fallacious) law $F(t) = m {dx\over dt}$. Later, time is reversed by "changing $t$ to $-t$" and the result is mentioned as $F(-t) = -m{dx\over dt}$. This made absolutely no sense to me. So I tried to formulate this in precise mathematical terms. The following is my result.

Let $\tau: \mathbb{R\to R}$ be a function defined by $\tau (t):=-t$ and let $\dot{f}$ denote the derivative of a differentiable function $f$. Let $F$ and $x$ be differentiable functions satisfying $F(t) = m\dot x(t)$ for some constant $m$. Then $F(-t) =-m\dot{(x\circ\tau)}(t)$.

Question: Am I correct?