How do you find the second derivative of a parametric function?

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Answer 1 · 2014-10-02T22:28:33

Let ${(x=x(t)),(y=y(t)):}$ .

First Derivative

${dy}/{dx}={{dy}/{dt}}/{{dx}/{dt}}={y'(t)}/{x'(t)}$

Second Derivative

${d^2y}/{dx^2}=d/{dx}[{y'(t)}/{x'(t)}]=1/{{dx}/{dt}}{d}/{dt}[{y'(t)}/{x'(t)}]$

$=1/{x'(t)}cdot{y''(t)x'(t)-y'(t)x''(t)}/{[x'(t)]^2}$

$={y''(t)x'(t)-y'(t)x''(t)}/{[x'(t)]^3}$

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