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}#

I hope that this was helpful.