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Math Help - implicit differentiation (finding the 1st and 2nd derivative)

  1. #1
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    implicit differentiation (finding the 1st and 2nd derivative)

    With these, i cannot get the answer that is said from the book.
    1. y^2 = e^(x^2) + 2x
    I already got the 1st one which is: (xe^(x^2) + 1) / y

    for the 2nd:
    ((y)(e^(x^2) + (2x^2)e^(x^2)) - (xe^(x^2) + 1)(y')) / y^2

    ((y)(e^(x^2) + (2x^2)e^(x^2)) - (xe^(x^2) + 1)((xe^(x^2) + 1) / y))
    / y^2

    Multiplied by y and got:

    (y^2*e^(x^2) ( 1+2x^2) - (xe^(x^2))(xe^(x^2) + 1)) / y^3 and this is

    where I totally got lost... but the book said the answer is

    (((2x^2)*(y^2) + y^2 - 2x)(e^(x^2)) - x^2e^(2x^2) - 1) / y^3

    2. 2 *sqrt(y) = x - y

    using the product rule:

    I got y^(-1/2) = 1 - y'

    I multiplied a sqrt(y) (to get rid of the fraction), but it didn't do me any good and eventually got

    completely confused. The answer says sqrt(y)/(sqrt(y) + 1) for the 1st

    and 1 / (2(sqrt(y) + 1) ^3 for the 2nd.
    Last edited by driver327; October 22nd 2009 at 02:07 PM. Reason: further reasoning
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  2. #2
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    2sqrt(y) = x - y implies 4y = x^2 - 2xy + y^2. Differentiating yields 4y' = 2x - (2y + 2xy') + 2yy'. Use algebra to solve for y'.
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  3. #3
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    Quote Originally Posted by rn443 View Post
    2sqrt(y) = x - y implies 4y = x^2 - 2xy + y^2. Differentiating yields 4y' = 2x - (2y + 2xy') + 2yy'. Use algebra to solve for y'.
    I am not exactly sure how you got 4y' from 2sqrt(y), but I tried this out again and got:

    2((1/2) (y^(-1/2)) (y') = 1- y'

    (y^(-1/2)) (y') = 1 - y'
    y' + (y^(-1/2)) (y') = 1
    y' (1+ (y^(-1/2))) = 1
    y' = 1 / (1+ (y^(-1/2)))
    Which i think amounts to:

    y' = ((y^(1/2)) + 1)

    What am I doing wrong here?
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  4. #4
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    Quote Originally Posted by driver327 View Post
    With these, i cannot get the answer that is said from the book.
    1. y^2 = e^(x^2) + 2x
    I already got the 1st one which is: (xe^(x^2) + 1) / y

    for the 2nd:
    ((y)(e^(x^2) + (2x^2)e^(x^2)) - (xe^(x^2) + 1)(y')) / y^2

    ((y)(e^(x^2) + (2x^2)e^(x^2)) - (xe^(x^2) + 1)((xe^(x^2) + 1) / y))
    / y^2

    Multiplied by y and got:

    (y^2*e^(x^2) ( 1+2x^2) - (xe^(x^2) + 1)(xe^(x^2) + 1)) / y^3 and this

    is where I totally got lost... but the book said the answer is

    (((2x^2)*(y^2) + y^2 - 2x)(e^(x^2)) - x^2e^(2x^2) - 1) / y^3
    I realized that i dropped the 1 (in blue) when i multiplied the y so i now get:

    (y^2*e^(x^2) ( 1+2x^2) - (xe^(x^2) + 1)^2) / y^3
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  5. #5
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    really need help on this 1...

    I think I fixed most of the errors on both problems... but somehow, i still get the wrong answer.
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