How do you twice differentiate #x^3+y^3=1#?

Answer 1

#y''=-(2x)/(y^5)#

#x^(3)+y^(3)=1#

We use implicit differentiation:

#D[x^(3)+y^(3)]=D[1]#

So:

#3x^2+3y^2y'=0#
#y'=-(cancel(3)x^2)/(cancel(3)y^2)=-(x^2)/(y^(2)# #color(red)((1))#
So now we have to differentiate again using the quotient rule. A useful tip is to start and finish with the function on the bottom #rArr#
#y''=(y^(2)(-2x)-(-x^(2)).2y.y')/(y^(4))# #color(red)((2))#
We already know from #color(red)((1))# that :
#y'=-x^(2)/y^2#
So we can substitute that expression for #y'# into #color(red)((2))# #rArr#
#y''=(y^(2)(-2x)-(-x^(2)).2y.((-x^(2))/(y^(2))))/(y^(4)#
#y''=[-2xy^2-(2x^4)/(y)]/[y^4]#
Multiplying top and bottom by #y# #rArr#
#y''=([-2xy^3-2x^4])/(y^5)#
#y''=(-2x[y^(3)+x^(3)])/(y^5)# #color(red)((3))#

Since we know that:

#x^3+y^3=1#
We can substitute that value of #1# into #color(red)((3))rArr#
#y''=(-2x)/(y^5)#
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Answer 2

To twice differentiate (x^3+y^3=1), you first need to express it in terms of one variable, and then differentiate twice with respect to that variable. Let's express it in terms of (y), and then differentiate twice with respect to (x).

  1. Solve for (y): [y^3 = 1 - x^3] [y = (1 - x^3)^{1/3}]

  2. Differentiate once with respect to (x): [\frac{dy}{dx} = \frac{d}{dx}(1 - x^3)^{1/3}]

  3. Differentiate twice with respect to (x): [\frac{d^2y}{dx^2} = \frac{d}{dx}\left(\frac{d}{dx}(1 - x^3)^{1/3}\right)]

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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