Question

Question #518f7

Answer 1

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

Explanation:

To find `(d^2y)/dx^2`, we must first find `dy/dx` which we can do using implicit differentiation.

`d/dx[x^2-y^2]=d/dx[5]`
`d/dxx^2-d/dxy^2=d/dx5` (Sum Rule)
`2x-2ydy/dx=0`
`2x=2ydy/dx`
`(2x)/(2y)=dy/dx`
`dy/dx=x/y`

Now, we can find the second derivative.

`d/dx[dy/dx]=d/dx[x/y]`
`(d^2y)/dx^2=(yd/dxx-xd/dxy)/y^2` (Quotient Rule)
`(d^2y)/dx^2=(y-xdy/dx)/y^2`
`(d^2y)/dx^2=(y-x(x/y))/y^2`
`(d^2y)/dx^2=(y-x^2/y)/y^2`
`(d^2y)/dx^2=(y^2-x^2)/y^3`