Let's say that I have some particle travelling along a 2D trajectory mapped out by some infinitely differentiable function $f$ (in this case $f(x) = x^{-1}$). I know the velocity of the particle solely in the $x$ dimension (i.e. $\frac{dx}{dt}$), but I want to find the particle's velocity along the trajectory. How might I go about doing this?
(Although I need to know this for a physics problem, I felt that the question actually fell more generally into the realm of mathematics. If others disagree and think this should be moved to the physics stack exchange site, feel free to comment below.)
Notice that the particle's velocity is a vector, and you should represent is as such. You know the graph of the particle's actual motion, which is a large piece of information you should take advantage of. Don't think of it as $f(x) = \frac{1}{x}$ - think of it as $y = \frac{1}{x}.$ Notice that the velocity vector is generally represented as $<\frac{dx}{dt},\frac{dy}{dt}>.$ You noted that you were given $\frac{dx}{dt},$ and by parametrics, you know that $\frac{dy}{dt} = \frac{dy}{dx}\frac{dx}{dt}.$ I think you can take it from here. Hope this helped!