Let $\displaystyle x(t)=2t^2+2$ and $\displaystyle y(t)=3t^4+4t^3$, then

find $\displaystyle \frac{d^2y}{dx^2}$ at point (8,80).

Thanks.

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- Nov 7th 2011, 12:07 AMchipaiChain rule of second order differential
Let $\displaystyle x(t)=2t^2+2$ and $\displaystyle y(t)=3t^4+4t^3$, then

find $\displaystyle \frac{d^2y}{dx^2}$ at point (8,80).

Thanks. - Nov 7th 2011, 02:52 AMsbhatnagarRe: Chain rule of second order differential
To find $\displaystyle \frac{d^2y}{dx^2}$, you must know y(x).

$\displaystyle x=2t^2+2 \Leftrightarrow t=\pm\sqrt{\frac{x-2}{2}}$.

Therefore, $\displaystyle y=3t^4+4t^3 \Leftrightarrow y=3(\frac{x-2}{2})^2\pm 4(\frac{x-2}{2})^{\frac{3}{2}}$

$\displaystyle y(x)=3(\frac{x-2}{2})^2\pm 4(\frac{x-2}{2})^{\frac{3}{2}}$

$\displaystyle y'(x)=?$

$\displaystyle y''(x)=?$