Question

How do you find the second derivative implicitly of `x^2 + xy = 5`?

Answer 1

`(d^2y) /dx^2 = 10/x^3`

Explanation:

Implicit differentiation first time is, `2x+xdy/dx+y=0`

Note that this gives `(dy)/(dx)=(-y-2x)/x`

Implicit differentiation second time gives,

`2+x(d^2y)/dx^2+dy/dx +dy/dx`=0. Hence,

`x(d^2y)/dx^2= -2-2 (dy/dx)=-2 -2/x (-y-2x)=2+2y/x=2+2(5/x^2 -1)= 10/x^2`

`(d^2y)/dx^2 = 10/x^3`