If a curve is represented parametrically by the equation `x=f(t) and y=g(t)" then prove that "(d^(2)y)/(dx^(2))=-[(g'(t))/(f'(t))]^(3)((d^(2)x)/(dy^(2)))`

Explanation

`(dx)/(dy)=(1)/((dy)/(dx))`
`rArr" "(d^(2)x)/(dy^(2))=(d)/(dy)((1)/((dy)/(dx)))=(d)/(dx)((1)/((dy)/(dx))).(dx)/(dy)=(-1)/(((dy)/(dx))^(2)).((d^(2)y)/(dx^(2)))((dx)/(dy))`
`therefore" "(d^(2)x)/(dy^(2))=((-d^(2)y)/(dx^(2)))/(((dy)/(d))^(3))or (d^(2)y)/(dx^(2))=-((dy)/(dx))^(3).(d^(2)x)/(dy^(2))`
`rArr" "(d^(2)y)/(dx^(2)=-((dy//dt)/(dx//dt))^(3).(d^(2)x)/(dy^(2))`
`rArr" "(d^(2)y)/(dx^(2))=-[(g'(t))/(f'(t))]^(-3)((d^(2)x)/(dy^(2)))`