Question:
If $x$ and $y$ are connected parametrically by the equation, without eliminating the parameter, find $\frac{d y}{d x}$.
$x=4 t, y=\frac{4}{t}$
Solution:
The given equations are $x=4 t$ and $y=\frac{4}{t}$
$\frac{d x}{d t}=\frac{d}{d t}(4 t)=4$
$\frac{d y}{d t}=\frac{d}{d t}\left(\frac{4}{t}\right)=4 \cdot \frac{d}{d t}\left(\frac{1}{t}\right)=4 \cdot\left(\frac{-1}{t^{2}}\right)=\frac{-4}{t^{2}}$
$\therefore \frac{d y}{d x}=\frac{\left(\frac{d y}{d t}\right)}{\left(\frac{d x}{d t}\right)}=\frac{\left(\frac{-4}{t^{2}}\right)}{4}=\frac{-1}{t^{2}}$
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