Find . `(d^2y)/(dx^2),ifx=a t^2,y=2a tdot`

Explanation

`x=at^(2),y=2at`
`therefore(dx)/(dt)=(d)/(dt)(at^(2))=a(d)/(dt)(t^(2))=axx2t=2at" ... (1)"`
and `(dy)/(dt)=(d)/(dt)(2at)=2a(d)/(dt)(t)`
`=2axx1=2a`
`therefore(dy)/(dx)=((dy//dt))/((dx//dt))=(2a)/(2at)=(1)/(t)`
`therefore(d^(2)y)/(dx^(2))=(d)/(dx)((1)/(t))=(d)/(dt)(t^(-1)).(dt)/(dx)`
`=(-1)t^(-2).(1)/(((dx)/(dt)))=(-1)/(t^(2))xx(1)/(2at)" ...[By (1)]"`
`=-(1)/(2at^(3))`