Let $x$ depend on $t$. $\dot{x}$ is derivative $x$ over $t$. I want to know if there are formulas to simplify $\frac{d\dot{x}}{dx}$? Any hint or thought is appreciated. Thank you!
how to calculate $\frac{d\dot{x}}{dx}$
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1@anon : I'm not the one who downvoted. It was already downvoted when I began writing my comment. I acknowledge, however, that I misunderstood capoluca's question and that I tried to figure out there is no such function $F$ satisfying $d\dot{x}/dx = F(x, \dot{x})$. – 2012-01-30
2 Answers
Put succinctly,
$\large \frac{\mathrm{d}\dot{x}}{\mathrm{d}x}=\frac{(\mathrm{d}\dot{x}/\mathrm{d}t)}{(\mathrm{d}x/\mathrm{d}t)}=\ddot{x}/\dot{x}.$
With more exposition: set $\Delta x=x(t+\Delta t)-x(t)$ and $\Delta \dot{x}=\dot{x}(t+\Delta t)-\dot{x}(t)$. Substitution gives
$\large\lim\limits_{\Delta x\to0}\frac{\Delta \dot{x}}{\Delta x}=\lim\limits_{\Delta t\to0}\frac{\Delta \dot{x}}{\Delta x}=\lim\limits_{\Delta t\to 0}\frac{\Delta\dot{x}/\Delta t}{\Delta x/\Delta t}=\ddot{x}/\dot{x}.$
The substitution is valid because $\Delta t\to0$ as we let $\Delta x\to0$.
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0Thanks, I like your explanation. – 2012-01-30
$\frac {d\dot{x}}{dx}=\frac{d}{dx}\left(\frac{dx}{dt}\right)=\frac{d}{dt}\left(\frac{dx}{dt}\right)\cdot \frac{dt}{dx}=\frac{d}{dt}(\dot{x})\cdot \frac{1}{\dot{x}}=\frac{\ddot{x}}{\dot{x}}$