In `(dy)/(dx),x` is a independent variable and `y` is the dependent variable. If independent and dependent variables are interchanged `(dy)/(dx)` becomes `(dx)/(dy)` and these two are connected by the relation `(dy)/(dx)dot(dx)/(dy)=1` . Find a relation between `(d^2y)/(dx^2)a n d(d^2x)/(dy^2)` .

Explanation

Given that,
`(dy)/(dx)dot(dx)/(dy)=1`
so,`(dx)/(dy)=1/((dy)/(dx))`
differentiate wrt `x` we get,
`=>(d^2x)/(dy^2)=(0-d/dy(dy)/(dx))/ ((dy)/(dx))^2`
`=>(d^2x)/(dy^2)=((-dx)/dyd/dx(dy)/(dx))/ ((dy)/(dx))^2`
`=>(d^2x)/(dy^2)=(-(d^2y)/(dx^2))/ ((dy)/(dx))^3`
which is the required relation between `(d^2y)/(dx^2)a n d(d^2x)/(dy^2)`
.