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September 1st, 2016, 05:05 AM  #1 
Member Joined: Mar 2015 From: uk Posts: 33 Thanks: 1  Chain rule and second order derivatives
I'm trying to work out the following question: Use the chain rule to show that (d2y/dx2).((dx/dy)^3) + d2x/dy2 = 0 I picked y=x^3 to convince myself that it is correct (for that equation) but don't know how to go about using the chain rule to prove it There is a hint saying: (dy/dx).(dx/dy) = 1 differentiate this Thanks 
September 1st, 2016, 05:14 AM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,655 Thanks: 2633 Math Focus: Mainly analysis and algebra 
What happens if you write $v(y)={\mathrm d x \mathrm d y}$, giving $y'v=1$? Differentiate implicitly.

September 1st, 2016, 04:15 PM  #3 
Banned Camp Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 125 
dy/dx=y' dx/dy=1/y' d$\displaystyle ^{2}$x/dy$\displaystyle ^{2}$=(1/y'$\displaystyle ^{2}$)y''(dx/dy)=(1/y'$\displaystyle ^{3}$)y'' Substituting into OP: y''(1/y'$\displaystyle ^{3}$)(1/y'$\displaystyle ^{3}$)y''=0 
September 2nd, 2016, 05:45 AM  #4 
Banned Camp Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 125  
September 10th, 2016, 03:13 PM  #5 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,655 Thanks: 2633 Math Focus: Mainly analysis and algebra 
Yes. There was quite obviously a typo in that post. Luckily the OP was able to see that, even if you were not.

September 10th, 2016, 06:50 PM  #6 
Banned Camp Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 125 
There was no indication anyone understood what you meant until after I responded.

September 10th, 2016, 07:47 PM  #7 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,655 Thanks: 2633 Math Focus: Mainly analysis and algebra 
You mean apart from the people who thanked me.

September 10th, 2016, 08:09 PM  #8 
Banned Camp Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 125  

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chain, derivatives, order, rule 
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