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Math Help - A matter regarding Leibniz notation

  1. #1
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    Question 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
    \frac{dy/dt}{dx/dt} = \frac{dy}{dx}
    or is this just the Leibniz notation playing tricks on us?

    Thanks in advance.
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  2. #2
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    Re: A matter regarding Leibniz notation

    Never mind. I answered my own question. You just take the chain rule
    \frac{dy}{dx} = \frac{dy}{dt} \frac{dt}{dx}
    and divide both sides by dt/dx.
    \frac{dy/dx}{dt/dx} = \frac{dy}{dt}
    Oh well, I'll be all alone here on this thread then.
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  3. #3
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    Re: A matter regarding Leibniz notation

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

    \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*}
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