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given-x-y-continuous-and-differenriable-such-that-x-x-t-y-y-t-does-d-2-y-dx-2-d-2-y-dt-2-d-2-x-dt-2-




Question Number 1996 by 123456 last updated on 28/Oct/15
given x,y continuous and differenriable  such that   { ((x=x(t))),((y=y(t))) :}  does  (d^2 y/dx^2 )=((d^2 y/dt^2 )/(d^2 x/dt^2 ))?
$$\mathrm{given}\:{x},{y}\:\mathrm{continuous}\:\mathrm{and}\:\mathrm{differenriable} \\ $$$$\mathrm{such}\:\mathrm{that} \\ $$$$\begin{cases}{{x}={x}\left({t}\right)}\\{{y}={y}\left({t}\right)}\end{cases} \\ $$$$\mathrm{does} \\ $$$$\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }=\frac{\frac{{d}^{\mathrm{2}} {y}}{{dt}^{\mathrm{2}} }}{\frac{{d}^{\mathrm{2}} {x}}{{dt}^{\mathrm{2}} }}? \\ $$
Answered by prakash jain last updated on 29/Oct/15
(dy/dx)=(dy/dt)∙(dt/dx)  (d^2 y/dx^2 )=((dt/dx))^2 ∙(d^2 y/dt^2 )+(dy/dt)∙(d^2 t/dx^2 )
$$\frac{{dy}}{{dx}}=\frac{{dy}}{{dt}}\centerdot\frac{{dt}}{{dx}} \\ $$$$\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }=\left(\frac{{dt}}{{dx}}\right)^{\mathrm{2}} \centerdot\frac{{d}^{\mathrm{2}} {y}}{{dt}^{\mathrm{2}} }+\frac{{dy}}{{dt}}\centerdot\frac{{d}^{\mathrm{2}} {t}}{{dx}^{\mathrm{2}} } \\ $$

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