Question

I need to know second order derivative `(d^2y)/dx^2` for `6xy^6-2y^(1/2)=52y^2` using implicit differentiation at point `(9,1)`?

I can get the first order differentiation just fine. I can't seem to get the right answer though

Answer 1

0.0142, to three significant figures

Explanation:

Equation:
`6xy^6-2y^(1/2)=52y^2`

Differentiate:
`6y^6+36xy^5dy/dx-y^(-1/2)dy/dx=104ydy/dx`

Collect `dy/dx` terms:
`dy/dx(104y-36xy^5+y^(-1/2))=6y^6`

Differentiate again:
`(d^2y)/(dx^2)(104y-36xy^5+y^(-1/2))+dy/dx(104dy/dx-36y^5-180xy^4dy/dx-1/2y^(-3/2)dy/dx)=36y^5dy/dx`

Collect `dy/dx` and `(dy/dx)^2` terms:
`(d^2y)/(dx^2)(104y-36xy^5+y^(-1/2))=72y^5dy/dx-(dy/dx)^2(104-180xy^4-1/2y^(-3/2))`

Expression for `dy/dx` from the working above:
`dy/dx=(6y^6)/(104y-36xy^5+y^(-1/2))`

Substitute this in to the expression for `(d^2y)/(dx^2)`:
`(d^2y)/(dx^2)(104y-36xy^5+y^(-1/2))=(432y^11)/(104y-36xy^5+y^(-1/2))-(36y^12)/(104y-36xy^5+y^(-1/2))^2(104-180xy^4-1/2y^(-3/2))`

Combine factors:
`(d^2y)/(dx^2)(104y-36xy^5+y^(-1/2))^2/(36y^11)=12-y/(104y-36xy^5+y^(-1/2))(104-180xy^4-1/2y^(-3/2))`

The question asks us to evaluate this expression at the specific point `(x,y)=(9,1)`, a point which we can verify quickly is a solution of the original equation.

So:
`((d^2y)/(dx^2))(9,1)*(104-324+1)^2/(36)=12-1/(104-324+1)(104-1620-1/2)`

`((d^2y)/(dx^2))(9,1)*(-219)^2/(36)=12-1/(-219)(3033/2)`

`((d^2y)/(dx^2))(9,1)=(12-1/(-219)(3033/2))/((-219)^2/36)`

`((d^2y)/(dx^2))(9,1)=(12+3033/438)/(73^2/4)`

`((d^2y)/(dx^2))(9,1)=(48+6066/219)/(73^2)`

This isn't a pleasant expression, so we'll simply evaluate it numerically:

0.0142, to three significant figures