Question

x^6 + y^6 = 1

What does:
y'' =

My book says y'' = (-5x^4) / (y^11)
But I don't understand how.

My work so far:

y' =

6x^(5) + 6y^(5) * dy/dx = 0

y' = (-x^5)/(y^5)

Now I use quotient rule. (f'g - fg')/(g^2)

y'' = [(-5x^4)(y^5) + (x^5)(5y^4)(dy/dx)] / (y^10)

Then i substituted (-x^5)/(y^5) for (dy/dx)

Then I simplified, and factored.

Ended up with (25x^10 - 5x^4) / (y^10)

Answer 1

If you need me to show more work let me know.

Answer 2

I've been watching the patrickJMT Calcus videos and they are pretty helpful.

Answer 3

Ima double check.

Answer 4

Hold on I'm gonna check this out. (getting pencil & paper)

Answer 5

yeah... your second derivative is incorrect...

Answer 6

this is what I got: (-5x^10 - 5x^4*y^6)/y^11

Answer 7

when you factor out (-5x^4) from that you'll end up with (x^6 + y^6) = 1

Answer 8

http://www.wolframalpha.com/input/?i=d^2%2Fdx^2+%281+-+x^6%29^%281%2F6%29

Answer 9

just solve for y and take two derivatives

Answer 10

there may be a point in there where you canepress a term in terms of y

Answer 11

Wolfram doesn't do implicit differentiation to the best of my knowledge.

Answer 12

ohhh haha im a dummy yea that way woud be way easier...

Answer 13

dplanc how did you get that.

Answer 14

Yeah I usually use wolfram. It's pretty good.

Answer 15

i just did exactly what you did... double check your second derivative...
btw... instead of using quotient rule, I used product rule: (-x^5)*(y^-5)

Answer 16

kk

Answer 17

Ok, I've done this problem twice now (double-checking each method), once using the quotient rule the other using the product rule to differentiate -x^5/y^5 (or (-x^5*y^(-5) if you like).

For both methods I obtain:

(-5x^4y^6 -5x^10)/y^11

Answer 18

Something happened:

y'' = [(-5x^4)(y^5) + (x^5)(5y^4)(dy/dx)] / (y^10)
Then i substituted (-x^5)/(y^5) for (dy/dx)
Then I simplified, and factored. <<<=== RIGHT HERE

:-p Either that or it is way past time for me to be doing math and I should be going to sleep.

Answer 19

ok =P ty I'ma stare really hard at this until I get it TT.

Answer 20

So let's take take the second term in the numerator:

+5x^5*y^4*(-x^5/y^5)
=
-x^10/y

Answer 21

Ok i get that one

Answer 22

Oh my goodness. I typed it all up and it had errors: This version is ok.

You should have the fraction simplified like this, multiply by y/y to remove the irritating y from the numerator's fraction...

\[\huge \frac{-5x^4y^5-\frac{5x^{10}}{y}}{y^{10}}\cdot\frac{y}{y}\]

And this simplifies to

\[\huge \frac{-5x^4y^6-5x^{10}}{y^{11}}\]

And that's it. I mean you can factor out a -5x^4 if you like.

Answer 23

Yeah I got that too at first but I thought I did something terribly wrong because my book definitely gives the wrong answer TT. Thank you very much for all the help =D.

Answer 24

No prob. :) Have a good evening!