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Question
Find \[\frac{dy}{dx}\] ,when \[x = \frac{e^t + e^{- t}}{2} \text{ and } y = \frac{e^t - e^{- t}}{2}\] ?
Solution
\[\text{ We have, x } = \frac{e^t + e^{- t}}{2} \text{ and } y = \frac{e^t - e^{- t}}{2}\]
\[\Rightarrow \frac{dx}{dt} = \frac{1}{2}\left[ \frac{d}{dt}\left( e^t \right) + \frac{d}{dt}\left( e^{- t} \right) \right] \text{ and } \frac{dy}{dt} = \frac{1}{2}\left[ \frac{d}{dt}\left( e^t \right) - \frac{d}{dt} e^{- t} \right]\]
\[ \Rightarrow \frac{dx}{dt} = \frac{1}{2}\left[ e^t + e^{- t} \frac{d}{dt}\left( - t \right) \right] \text{ and } \frac{dy}{dt} = \frac{1}{2}\left[ e^t - e^{- t} \frac{d}{dt}\left( e^{- t} \right) \right]\]
\[ \Rightarrow \frac{dx}{dt} = \frac{1}{2}\left( e^t - e^{- t} \right) = y \text{ and } \frac{dy}{dt} = \frac{1}{2}\left( e^t + e^{- t} \right) = x \]
\[ \therefore \frac{dy}{dt} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{x}{y}\]
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