Question

differentiation   x=at^2  
                      y=2at

Answer 1

$x=a t^{2}$

Differenciating x with respect to t, we get

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

$y=2at$

Differentiating y with respect to t, we get

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

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

Answer 2

We can get dy/dx in many ways.

$x = at^2,\ \ \ \ y=2at\\ \\x=a(y/2a)^2=\frac{y^2}{4a}\\ \\y^2=4ax\\ \\Differentiating\ \ wrt\ \ x\\ \\2y. \frac{dy}{dx}= 4a\\ \\\frac{dy}{dx}=\frac{2a}{y}=\frac{2a}{2at}=\frac{1}{t}$