Expert: Abe Mantell - 5/28/2009

Question
How do I do this problem:
ln(8y)=4xy

Part a)
Use implicit differentiation to find the first derivative. dy/dx= ?

part b)
use Use implicit differentiation to find the second  derivative. d^2y/dx^2

part c)
find d^2y/dx^2 = 0
at (x,y).


no examples were taught in class, so I'm stuck.

Answer
Hello Brooke,

a) differentiating with respect to x gives:
 (1/8y)*8=4y+4x*y', by the product rule
= 1/y=4y+4x*y', now solve for y'
==> 4x*y'=1/y - 4y
==> y'=1/(4xy) - y/x = (4xy)^(-1) - y/x

b) differentiate again:
==> y''=-1*(4xy)^(-2) * (4y + 4xy') - (xy'-y)/x^2
==> y''=(4y+4xy')/(16x^2 y^2) - (xy'-y)/x^2
==>  y''=16*y^3*(-3+8*x*y)/(-1+4*x*y)^3

Setting y''=0 gives 16*y^3*(-3*8xy)=0
So, either y=0 (not possible), or xy=3/8 ==> y=3/(8x)
Then putting that into the original equation gives x=3e^(-3/2)
Which then gives y=(1/8)e^(3/2)


I hope this helps!

Abe