Question

If $x$ and $y$ are connected parametrically by the given equation, then without eliminating the parameter, find $\frac{dy}{dx}$.
$x=4t,y=\frac{4}{t}$

Answer 1

We have $x=4t,y=\frac{4}{t}$
$\Rightarrow \frac{dx}{dt}=\frac{d}{dt}\left(4t\right)=4$
and $\frac{dy}{dt}=\frac{d}{dt}\left(\frac{4}{t}\right)=4.\frac{d}{dt}\left(\frac{1}{t}\right)=4.\left(\frac{-1}{t^2}\right)=\frac{-4}{t^2}$
$\therefore \frac{dy}{dx}=\frac{\left(\frac{dy}{dt}\right)}{\left(\frac{dx}{dt}\right)}=\frac{\frac{-4}{t^2}}{4}=\frac{-1}{t^2}$