Question

find dy/dx if x = at^2 , y = 2at

Answer 1

Answer:

$\dfrac{dy}{dx}=\dfrac{y}{2x}$

Step-by-step explanation:

Given,

$x=at^2\\y=2at$

For finding$\dfrac{dy}{dx}$ first we find the derivative of$y$ with respect to$t$ and the derivative of$x$ with respect to$t$.

$\dfrac{dx}{dt}=\dfrac{d}{dt}\left[at^2\right]\\\dfrac{dx}{dt}=2at$

and

$\dfrac{dy}{dt}=\dfrac{d}{dt}\left[2at\right]\\\dfrac{dx}{dt}=2a$.

Now we find the fraction of the derivative of y with the derivative of x:

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

So we have the solution:-

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

Now, we convert the solution in$x$ and$y$ variable.

We have,

$\dfrac{y}{x}=\dfrac{2at}{at^2}\\\dfrac{y}{x} = \dfrac{2}{t}$

So, we have

$\dfrac{1}{t} = \dfrac{y}{2x}$

Finally we have the solution,

$\dfrac{dy}{dx} = \dfrac{y}{2x}$

Answer 2

Given : y = at²  ,   x =  at

To Find : dy/dx

Solution:

y = at²

dy/dt  = 2at

x =  at

dx/dt  = a

(dy/dx)  = (dy/dt )/  (dx/dt)  =  2at/a

=> dy/dx  = 2t

Another method:

y = at²

=> dy/dx  = 2at. dt/dx

x =  at

=> t = x/a

=> dt/dx  = 1/a

Substitute  dt/dx= 1/a  in   dy/dx  = 2at. dt/dx

=> dy/dx  = 2at. (1/a)

=>  dy/dx  = 2t  

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