Question

if x=1/t and y=1-1/t find dy/dx

Answer 1

➫Question :-

$\tt \: if \: \: x = \frac{1}{t} \: \: and \: y = 1 - \frac{1}{t} \: \: then \: find \\ \tt \frac{dy}{dx}$

➫Answer :-

$\to \large \boxed{ \tt \frac{dy}{dx} = - 1} \:$

➫Solution :-

We have ,

$\to \tt x = \frac{1}{t} \\ \\ \red{\sf differentiate \: with \: respect \: to \: t \: } \\ \\ \to \tt \frac{dx}{dt} = \frac{d}{dt} \frac{1}{t} \\ \\ \to \tt \frac{dx}{dt} = log(t) \: \: ......\: eq.(1) \\ \\ \bf \: \: \green{and \: we \: have \: } \\ \\ \to \tt y = 1 - \frac{1}{t} \\ \\ \sf \pink{differentiate \: with \: respect \: to \: t} \\ \\ \to \tt \frac{dy}{dt} = 0 - \frac{d}{dt} \: \frac{1}{t} \\ \\ \to \tt \frac{dy}{dt} = - log(t) \: ...... \: eq.(2) \\ \\ \sf \: Apply \implies \tt \frac{eq.(2)}{eq.(1)} \\ \\ \to \tt \frac{ \frac{dy}{dt} }{ \frac{dx}{dt} } = \frac{ log(t) }{ - log(t) } \\ \\ \to \tt \frac{dy}{dt} \times \frac{dt}{dx} = - 1 \\ \\ \to \boxed{ \tt \frac{dy}{dx} = - 1}$