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Let $x(t) = 2t^2 + 2$, $y(t) = 3t^4 + 4t^3$. Find $$\frac{d^2y}{dx^2}.$$

My question is:

  1. Is this parameter function?
  2. I have tried to find the relationship between $x$ and $y$, but it seems this does not work. Is there any trick for such problem?
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@Marvis I'm sorry to make a type error. I have edited "$4t^3$" instead of "$4^3$" – John Hass Oct 27 '12 at 18:26
up vote 4 down vote accepted

$$\dfrac{dy}{dx} = \dfrac{dy}{dt} \dfrac{dt}{dx}$$ $$\dfrac{d^2 y}{dx^2} = \dfrac{d\left(\dfrac{dy}{dt} \dfrac{dt}{dx} \right)}{dt} \dfrac{dt}{dx}$$ Or in your case, you could relate $y$ and $x$ directly as $$y = 3 \left(\dfrac{x-2}2 \right)^2 + 4\left(\dfrac{x-2}2 \right)^{3/2}$$

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I'm sorry to make a type error. I have edit "$4t^3$" instead of "$4^3$" – John Hass Oct 27 '12 at 18:25

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