Implicit differentiation problem

Limits, differentiation, related rates, integration, trig integrals, etc.
Andromeda
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Implicit differentiation problem

Postby Andromeda » Tue Nov 23, 2010 6:37 pm

Find y" by implicit differentiation

x^3 + y^3 = 1

Here's my work:
3x^2 + 3y^2y' = 0

y' = (-3x^2)/(3y^2) = (-x^2)/(y^2)

y" = (-2xy^2 + 2x^2y(dy/dx)) / y^4

= 2(-xy - x^4y^-2) / y^3

The back of the book says that the answer is -2x / y^5. I don't understand how they're simplifying it to that. Thanks for any help.

nona.m.nona
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Re: Implicit differentiation problem

Postby nona.m.nona » Tue Nov 23, 2010 9:06 pm

They appear to have used a couple substitutions and then simplified. Your answer is equivalent.

Andromeda wrote:y" = (-2xy^2 + 2x^2y(dy/dx)) / y^4

This is correct. However, try plugging the fractional form for dy/dx into the numerator, and simplifying:

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

Plug this back into the expression for y":

. . . . .

. . . . .

Now factor:

. . . . .

Now substitute:

. . . . .

Andromeda
Posts: 11
Joined: Tue Oct 12, 2010 1:56 am
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Re: Implicit differentiation problem

Postby Andromeda » Wed Nov 24, 2010 1:22 am

Ah I see. Thank you.


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