Question

Find dy/dx when x=2at and y=2at^2

Answer 1

Answer:

$\boxed{\bold{\huge{\pink{2 \: t}}}}$

Step-by-step explanation:

$\bold{\underline{\orange{Given}}}\longrightarrow \\ x \: = 2at \: \: and \: y = 2a {t}^{2}$

$\bold{\underline{\red{To \: find}}} \\ value \: of \: \dfrac{dy}{dx}$

$\bold{\underline{\blue{Concept \: used}}}\longrightarrow \\ 1) \frac{d}{dx}( {x}^{n} ) = n \: x ^{n - 1} \\ 2) \dfrac{dy}{dx} = \dfrac{ \dfrac{dy}{dt} }{ \dfrac{dx}{dt} }$

$\bold{\underline{\green{Solution}}}\longrightarrow \\ x \: = 2at \\ differentiating \: with \: respect \: to \: t \\ \dfrac{dx}{dt} = \dfrac{d}{dt} (2at) \\ \dfrac{dx}{dt} = 2a \: \dfrac{d}{dt} (t) \\ \dfrac{dx}{dt} = 2a \: (1) \\ \dfrac{dx}{dt} = 2a$

$y \: = 4at \\ differentiating \: with \: respect \: to \: t \\ \dfrac{dy}{dt} = \dfrac{d}{dt} (2a {t}^{2} ) \\ \dfrac{dy}{dt} = 2a \dfrac{d}{dt} ( {t}^{2} ) \\ \dfrac{dy}{dt} =2 a \: (2 {t}^{2 - 1} ) \\ \dfrac{dy}{dt} = 4at$

$now$

$\dfrac{dy}{dx} = \dfrac{ \dfrac{dy}{dt} }{ \dfrac{dx}{dt} } \\ \dfrac{dy}{dx} = \dfrac{4at}{2a} \\ \dfrac{dy}{dx} = 2t$

Answer 2

given

$x = 2at \: \: \: and \: \: y = 2a {t}^{2}$

we have to find dy/dx

Now

$x = 2at \\ \\ \frac{dx}{dt} = 2a \frac{d}{dt} t \\ \\ \frac{dx}{dt} = 2a$

And

$y = 2a {t}^{2} \\ \\ \frac{dy}{dt} = 2a.2t \\ \\ \frac{dy}{dt} = 4at$

We can write

$\frac{dy}{dx} = \frac{dy}{dt} \times \frac{dt}{dx} \\ \\ \frac{dy}{dx} = 4at \times \frac{1}{2a} \\ \\ \frac{dy}{dx} = 2t$