So the chain rule for second derivatives is $$ \frac {d^2 y} {d t^2} = \frac{d}{dx}(\frac {dy} {dx}) \cdot \frac {dx} {dt} \cdot \frac {dx} {dt} + \frac {dy} {dx} \cdot \frac {d^2 x} {d t^2} = \frac{d^2 y}{d x^2} \cdot (\frac {dx} {dt})^2 + \frac {dy} {dx} \cdot \frac {d^2 x} {d t^2}$$
Today I came across this equation in a graphics/computer modeling course
$$\ddot C = \frac {d\dot C} {d \mathbf x} \cdot \dot {\mathbf x} + \frac {dC} {d \mathbf x} \cdot \ddot {\mathbf x}$$
Now what i would infer from this is that
$$\frac {d} {dx}(\frac {dy} {dx}) \cdot \frac {dx} {dt} = \frac {d} {dx}(\frac {dy} {dx} \cdot \frac {dx} {dt}) =\frac {d\dot y} { dx}$$
This sounds right but can someone point me to a rule or theorem that suggests that ? can the $dx/dt$ be moved in and out of the $d()/dt$ operator ??