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

1 Answer
Jun 16, 2018

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