Parametric derivatives of curves in three or more dimensions

I want to model a number of points in multidimensional (e.g. three dimensional) space by a one dimensional parametric curve. My ansatz was to take cubic spline interpolation functions between each of the given points in each dimension. Given the example of three dimensions, I have for each section of the curve the following equation: $$\mathbf{q}(t)= \left( \begin{array}{c} x(t)\\ y(t)\\ z(t) \end{array} \right)$$ This works quite well already; a smooth interpolation is done. However, for my specific application (theoretical chemistry and modeling of reaction surfaces), I need to know the partial derivatives of the first variable with respect to the second and third one, i.e.: $$\frac{\partial x(t)}{\partial y(t)} , \frac{\partial x(t)}{\partial z(t)}$$ (In my application, the x coordinate is the energy, and y and z are coordinates of the system). At the current stage, I do these derivatives numerically (finite differences), but I would like to have the analytical ones, too, which would be faster and more elegant. However, if I simply try to take the approach for two-dimensional parametric curves $\frac{dx}{dy}=\frac{\frac{dx}{dt}}{\frac{dy}{dt}}$, I would have: $$\frac{\partial x(t)}{\partial y(t)} = \frac{\frac{\partial x(t)}{\partial t}}{\frac{\partial y(t)}{\partial t}}, \frac{\partial x(t)}{\partial z(t)} = \frac{\frac{\partial x(t)}{\partial t}}{\frac{\partial z(t)}{\partial t}}$$ Unfortunately, this approach seems to be wrong, the results I get are quite horrible.. I already searched thoroughly in the web, but I found only the two dimensinal (x,y) case. Is there some kind of higher-dimensional generalization of the two dimensional case or is it simply not possible to get a analytical solution for that? Thank you in advance!

• Welcome! In your $3$-$D$ case, are $x,y,z$ independent variables. – Joe Sep 17 '18 at 1:28