If $x$ and $y$ are connected parametrically by the given equation, then without eliminating the parameter, find $\frac{d y}{d x}$
$x = 2 a t^{2}, y = a t^{4}$

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Solution

The given equations are $x = 2 a t^{2}, y = a t^{4}$
Then, $\frac{d x}{d t} = \frac{d}{d t} (2 a t^{2}) = 2 a . \frac{d}{d t} (t)^{2} = 2 a . 2 t = 4 a t$
and $\frac{d y}{d t} = \frac{d}{d t} (a t^{4}) = a . \frac{d}{d t} (t^{4}) = a . 4 . t^{3} = 4 a t^{3}$
$∴ \frac{d y}{d t} = \frac{(\frac{d y}{d t})}{(\frac{d x}{d t})} = \frac{4 a t^{3}}{4 a t} = t^{2}$

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