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