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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

$$\ddot C = \frac {d\dot C} {dx} \cdot \dot x + \frac {dC} {dx} \cdot \ddot x$$

Now what i would infer from this is that

$$\frac {d} {dx}(\frac {dy} {dx}) \cdot \frac {dx} {dt} = \frac {d\dot y} { dx}$$

This sounds right but can someone point me to a rule or theorem that suggests that. or a proof maybe ?

Today I came across this equation in a graphics/computer modeling course

$$\ddot C = \frac {d\dot C} {dx} \cdot \dot x + \frac {dC} {dx} \cdot \ddot x$$

Now what i would infer from this is that

$$\frac {d} {dx}(\frac {dy} {dx}) \cdot \frac {dx} {dt} = \frac {d\dot y} { dx}$$

This sounds right but can someone point me to a rule or theorem that suggests that. or a proof maybe ?

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