A matter regarding Leibniz notation

Dear mathematicians out there,

This is just a clarification regarding an identity I encountered that I couldn't find elsewhere on the Internet. In general, is it true that

$\displaystyle \frac{dy/dt}{dx/dt} = \frac{dy}{dx}$

or is this just the Leibniz notation playing tricks on us?

Thanks in advance.

Re: A matter regarding Leibniz notation

*Never mind.* I answered my own question. You just take the chain rule

$\displaystyle \frac{dy}{dx} = \frac{dy}{dt} \frac{dt}{dx}$

and divide both sides by dt/dx.

$\displaystyle \frac{dy/dx}{dt/dx} = \frac{dy}{dt}$

Oh well, I'll be all alone here on this thread then.

Re: A matter regarding Leibniz notation

Yes, it's true. You need to remember that $\displaystyle \displaystyle \begin{align*} \frac{1}{\frac{dY}{dX}} = \frac{dX}{dY} \end{align*}$. Using this property we have

$\displaystyle \displaystyle \begin{align*} \frac{\frac{dy}{dt}}{\frac{dx}{dt}} &= \frac{dy}{dt} \cdot \frac{1}{\frac{dx}{dt}} \\ &= \frac{dy}{dt} \cdot \frac{dt}{dx} \\ &= \frac{dy}{dx} \textrm{ by the Chain Rule} \end{align*}$