Question

If $x=a{t}^2,y=2at$, then $\frac{dy}{dx}=$_____;$t\ne 0$

• A. $at$
• B. $\frac{t}{2}$
• C. $\frac{2}{t}$
• D. $\frac{1}{t}$

Answer 1

Answer: (D)

$\frac{1}{t}$

$x=a{t}^2$

Differentiating both sides with respect to t

$\frac{dx}{dt}=\frac{d}{dt}\left(a{t}^2\right)$

$\frac{dx}{dt}=a\frac{d}{dt}\left(t^2\right)$

$\frac{dx}{dt}=2at$ (1)

$y=2at$

Differentiating both sides with respect to t

$\frac{dy}{dt}=\frac{d}{dt}\left(2at\right)$

$\frac{dy}{dt}=2a\frac{d}{dt}\left(t\right)$

$\frac{dy}{dt}=2a$ (2)

Now

$\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}$

From (1) and (2)

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