If x and y are connected parametrically by the equation,

Question:

If $x$ and $y$ are connected parametrically by the equation, without eliminating the parameter, find $\frac{d y}{d x}$.

$x=2 a t^{2}, y=a t^{4}$

Solution:

The given equations are $x=2 a t^{2}$ and $y=a t^{4}$

Then, $\frac{d x}{d t}=\frac{d}{d t}\left(2 a t^{2}\right)=2 a \cdot \frac{d}{d t}\left(t^{2}\right)=2 a \cdot 2 t=4 a t$

$\frac{d y}{d t}=\frac{d}{d t}\left(a t^{4}\right)=a \cdot \frac{d}{d t}\left(t^{4}\right)=a \cdot 4 \cdot t^{3}=4 a t^{3}$

$\therefore \frac{d y}{d x}=\frac{\left(\frac{d y}{d t}\right)}{\left(\frac{d x}{d t}\right)}=\frac{4 a t^{3}}{4 a t}=t^{2}$

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