I need to know second order derivative d2ydx2 for 6xy62y12=52y2 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:
6xy62y12=52y2

Differentiate:
6y6+36xy5dydxy12dydx=104ydydx

Collect dydx terms:
dydx(104y36xy5+y12)=6y6

Differentiate again:
d2ydx2(104y36xy5+y12)+dydx(104dydx36y5180xy4dydx12y32dydx)=36y5dydx

Collect dydx and (dydx)2 terms:
d2ydx2(104y36xy5+y12)=72y5dydx(dydx)2(104180xy412y32)

Expression for dydx from the working above:
dydx=6y6104y36xy5+y12

Substitute this in to the expression for d2ydx2:
d2ydx2(104y36xy5+y12)=432y11104y36xy5+y1236y12(104y36xy5+y12)2(104180xy412y32)

Combine factors:
d2ydx2(104y36xy5+y12)236y11=12y104y36xy5+y12(104180xy412y32)

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:
(d2ydx2)(9,1)(104324+1)236=121104324+1(104162012)

(d2ydx2)(9,1)(219)236=121219(30332)

(d2ydx2)(9,1)=121219(30332)(219)236

(d2ydx2)(9,1)=12+30334387324

(d2ydx2)(9,1)=48+6066219732

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

0.0142, to three significant figures