I've got a function $f(x(t),y(t),z(t),t)=0$ and want to calculate a double derivative over t.
For the first derivative I received:
${df \over dt}={\partial f \over \partial x}{\partial x \over \partial t}+{\partial f \over \partial y}{\partial y \over \partial t}+{\partial f \over \partial z}{\partial z \over \partial t}+{\partial f \over \partial t}=0$
For the second derivative, I have to calculate:
${d^2f \over dt^2}={\partial \over \partial t}\left ({\partial f \over \partial x}{\partial x \over \partial t}+{\partial f \over \partial y}{\partial y \over \partial t}+{\partial f \over \partial z}{\partial z \over \partial t}+{\partial f \over \partial t} \right )=0$
Did I calculate the first part correctly, because I think I lost sth?
${\partial \over \partial t}\left ({\partial f \over \partial x}{\partial x \over \partial t} \right )=\left ({\partial^2 f \over \partial x \partial x}{\partial x \over \partial t}+{\partial^2 f \over \partial x \partial y}{\partial y \over \partial t}+{\partial^2 f \over \partial x \partial z}{\partial z \over \partial t} \right ){\partial x \over \partial t}+{\partial f \over \partial x}{\partial^2 x \over \partial t^2}$