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

\[\Large y" = \frac{ [(-5x^{4}y^{5} + x^{5}(5y^{4})(-x^{5}/y^{5})]}{ y{10} }\]
Multiply top and bottom by y so the y^5 at the bottom in the second part of the numerator will cancel out. Also factor out -5x^4 in the numerator:

Answer 2

\[\Large y'' = \frac{ -5x^{4}(y^{6} + x^{6}) }{ y^{11} }\]

Answer 3

can we do the product rule of -x^5/y^5

Answer 4

But x^6 + y^6 = 1
So y'' = -5x^4 / y^11

Answer 5

ohhhhhhh gotcha but i can do the quotient rule right haha

Answer 6

cant

Answer 7

You don't have to take the derivative anymore. You are done with the derivative. All you have to do is to simplify. I just took what you had for the second derivative and I just did the simplification part.

Answer 8

You had these two lines:
y'' = [(-5x^4)(y^5) + (x^5)(5y^4)(dy/dx)] / (y^10)
Then i substituted (-x^5)/(y^5) for (dy/dx)

I did the substitution and simplified.

Answer 9

okay thank you !:)

Answer 10

you are welcome. Glad to see you did most of the work and just wanted some clarification for the last step.

Answer 11

wait i dont understand the multiplying the y to the numerator part

Answer 12

Let us start with the second part of the numerator: x^5(5y^4)(−x^5/y^5

Answer 13

Simplify it. -5x^10y^4/y^5 = -5x^10/y

Answer 14

oohhhhhhhhhh haha wow thank youuuu

Answer 15

np. yw.

Answer 16

are you a math teacher haha

Answer 17

not really. Software Engineer.

Answer 18

nice my dads an ee haha

Answer 19

oh cool. He will be able to easily help you out.

Answer 20

yeah exactly but hes sleeping cause he has to work and im always stuck on the same problem for hours

Answer 21

yeah work keeps people busy and tired when they return home. But it looks like you will do very well in calc. You did most of the work yourself and correctly.

Answer 22

Do well in your test. c ya.

Answer 23

ok one more question i factored it out and it was -5x^4(y^5+x^6)

Answer 24

(-5x^4)(y^5) -5x^10/y = -5x^4(y^5 + x^6/y) = -5x^4(y^6 + x^6) / y

Answer 25

and then you just multiply it by y/y?

Answer 26

How do you add y^5 + x^6/y ? Multiply each term by y and then divide by y.
(y^6 + x^6) / y

Answer 27

Or if you prefer: y^5 + x^6/y Multiply the first term by y/y: y^6/y + x^6/y and you can make y the common denominator: (y^6 + x^6) / y. That y will drop down to the denominator which already has a y^10 make it a y^11.

Answer 28

thank you. this helped me alot

Answer 29

you are welcome.