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If $x=t^{2}, y=t^{3}$, then what is $\frac{d^{2} y}{d x^{2}}$ equal to?
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Verified Answer
The correct answer is:
$\frac{3}{4 t}$
Let $x=t^{2}$ and $y=t^{3}$
$\Rightarrow \frac{d x}{d t}=2 t$ and $\frac{d y}{d t}=3 t^{2}$
$\therefore \quad \frac{d y}{d x}=\frac{d y / d t}{d x / d t}=\frac{3 t^{2}}{2 t}=\frac{3}{2} t$
$\Rightarrow \frac{d^{2} y}{d x^{2}}=\frac{3}{2} \cdot \frac{d t}{d x}=\frac{3}{2} \cdot \frac{1}{2 t}\left(\because \frac{d x}{d t}=2 t\right)$
$=\frac{3}{4 t}$
$\Rightarrow \frac{d x}{d t}=2 t$ and $\frac{d y}{d t}=3 t^{2}$
$\therefore \quad \frac{d y}{d x}=\frac{d y / d t}{d x / d t}=\frac{3 t^{2}}{2 t}=\frac{3}{2} t$
$\Rightarrow \frac{d^{2} y}{d x^{2}}=\frac{3}{2} \cdot \frac{d t}{d x}=\frac{3}{2} \cdot \frac{1}{2 t}\left(\because \frac{d x}{d t}=2 t\right)$
$=\frac{3}{4 t}$
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