From http://www.mathpages.com/home/kmath523/kmath523.htm
Variations in $x,y,z$ and $X$ at constant $t$ are independent of $t$ (since each of these variables is strictly a function of $t$), so we have $${\frac{\partial x}{\partial X}=\frac{\partial{\dot x}}{\partial{\dot X}}}$$
(This is just after equation (5) on the page.)
I'm having trouble making sense of this. If each of these variables is strictly a function of $t$, and $t$ is held constant, how does a time derivative (${\dot x}$) make sense?
If it were to mean that $t$ is replaced with a constant after the differentiation, then I could take the example of
$$ x(y)=X^{3/4}=t^{3}$$ $$ X(t)=t^{4}$$
I could then calculate
$$ {\frac{\partial x}{\partial X}=\frac{\dot x}{\dot X} = \frac{3}{4t}}$$
and
$$ {\frac{\partial {\dot x}}{\partial {\dot X}}=\frac{\ddot x}{\ddot X}=\frac{1}{2t}}$$
...and those are not equal for any $t$.
What is the justification for this step?
Note: This is a follow up to my first question about this, but that error was resolved. Rates of change for functions dependent on same variable