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Implicit Differentiation

  1. Sep 21, 2009 #1
    1. The problem statement, all variables and given/known data
    Use implicit differentiation to find ∂z/∂x and ∂z/∂y
    yz = ln(x + z)


    3. The attempt at a solution
    I came up with
    (x+2)/(x+2)(1-xy-yz)

    Could someone please help me solve this. I know to treat y as a constant and to multiply all the derivatives of z by ∂z/∂x
     
  2. jcsd
  3. Sep 21, 2009 #2

    rock.freak667

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    if y=constant

    the left side is just y(∂z/∂x)


    so now for ln(x+z) what happens when you differentiate this w.r.t to x ?
     
  4. Sep 21, 2009 #3
    well I get 1/(x+z)(1+(∂z/∂x)). But this is what I did when I got the incorrect answer.
     
  5. Sep 21, 2009 #4

    Dick

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    That's fine for the right side if you are doing d/dx. What's the left side? Isn't it (dy/dx)*z+y*(dz/dx)? I don't understand how your answer doesn't contain any derivatives.
     
  6. Sep 21, 2009 #5
    Because thats what im trying find. You differentiate z with respect to x.
    y(∂z/∂x)=1/(x+z)(1+(∂z/∂x)) is what I got but I dont think its right and if it is I messed something up when I solved for (∂z/∂x)
     
  7. Sep 21, 2009 #6

    Dick

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    You can't eliminate all of the derivatives from the solution of either dz/dx or dz/dy. Each solution has to contain the partial derivative of z wrt to the other variable.
     
  8. Sep 21, 2009 #7
    there are two different answers. the answer i got was just for ∂z/∂x. can someone please tell me if I did it right or not.
     
  9. Sep 21, 2009 #8

    Dick

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    No. You didn't do it right. If you are solving for dz/dx how can you get rid of dy/dx?
     
  10. Sep 21, 2009 #9
    y is a constant when you solve implicitly for (∂z/∂x)
     
  11. Sep 22, 2009 #10

    Dick

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    Of course it is. Sorry. I wasn't thinking. y(∂z/∂x)=(1/(x+z))*(1+(∂z/∂x)) is fine for a start. Now what do you get when you solve for ∂z/∂x?
     
  12. Sep 22, 2009 #11
    I figured it out. Thanks though. I got the wrong answer because I did the algebra wrong. I just assumed I did the calculus wrong. Thanks again.
     
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