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In (dy)/(dx),\ x is independent variable...

In `(dy)/(dx),\ x` is 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)*(dx)/(dy)=1` . Find a relation between `(d^2y)/(dx^2)` and `(d^2x)/(dy^2)`

Text Solution

Verified by Experts

Given that,

`(dy)/(dx)*(dx)/(dy)=1`

`(dx)/(dy)=(1)/((dx)/(dy))`

Differentiate w.r.t. `x`, we get

`(d^2x)/(dy^2)=(0-(d)/(dy)*(dy)/(dx))/((dy)/(dx))^2`

`(d^2x)/(dy^2)=(-(dx)/(dy)*(d)/(dz)*(dy)/(dz))/((dy)/(dx))^2`

`(d^2x)/(dy^2)=(-(d^2y)/(dx^2))/((dy)/(dx))^3`

Thus, above equation shows relation between `(d^2x)/(dy^2)` and `(d^2y)/(dx^2)`
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