Question

Find dy÷dx if x=at^2,y=2at
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Answer 1

$\text{It is given that }x = at^{2} \text{ and }y = 2at$

$\text{We must find }\dfrac{dy}{dx}$

$\implies \text{We can write }\dfrac{dy}{dx} \text{ as }\dfrac{\dfrac{dy}{dt}}{\dfrac{dx}{dt}}$

$\implies x = at^{2}$

$\text{Applying derivative in terms of }t$

$\implies \dfrac{dx}{dt} = \dfrac{d(at^{2})}{dt}$

$\implies \dfrac{dx}{dt} = a\times \dfrac{d(t^{2})}{dt}$

$\implies \dfrac{dx}{dt} = a\times2t$

$\implies \dfrac{dx}{dt} = 2at \text{ ------ (i)}$

$\implies y = 2at$

$\text{Applying derivative in terms of }t$

$\implies \dfrac{dy}{dt} = \dfrac{d(2at)}{dt}$

$\implies \dfrac{dy}{dt} = 2a\times \dfrac{dt}{dt}$

$\implies \dfrac{dy}{dt} = 2a \text{ ------ (ii)}$

$\implies \dfrac{dy}{dx} = \dfrac{\dfrac{dy}{dt}}{\dfrac{dx}{dt}}$

$\text{From (i) and (ii)}$

$\implies \dfrac{dy}{dx} = \dfrac{2a}{2at}$

$\implies \boxed{\dfrac{dy}{dx} = \dfrac{1}{t}}$