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Q.
If $x=t^2$, $y=t^3$, then $\frac{d^{2}\,y}{dx^{2}}$ is
Continuity and Differentiability
Solution:
$x=t^{2}\quad\ldots\left(i\right)$,
$y=t^{3}\quad\ldots\left(ii\right)$
From $(i)$ and $(ii)$, we get $y = x^{3/2}$
Differentiating w.r.t. to $x$, we get
$\frac{dy}{dx}=\frac{3}{2}x^{\frac{1}{2}}$
Again differentiating w.r.t. to $x$, we get
$\frac{d^{2}\,y}{dx^{2}}=\frac{3}{4\sqrt{x}}=\frac{3}{4t}$ (from eqn. $(i)$)