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Multivariable Chain-Rule Problem

  1. Jun 6, 2016 #1

    S.R

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    1. The problem statement, all variables and given/known data
    Let g(x, y) = f(sin(y), cos(x)). Find the second partial derivative of g with respect to x (g_xx).

    2. Relevant equations


    3. The attempt at a solution
    I attempted to find g_x, but I'm not entirely sure how chain rule applies in this situation.

    Is this correct?

    g_x = f_x(sin(y), cos(x)) * (-sin(x))
     
  2. jcsd
  3. Jun 6, 2016 #2
    To get the derivative with respect to ##x,## by chain rule: $$\frac{\partial g}{\partial x}=\frac{\partial f}{\partial x}\frac{\partial \sin y}{\partial x}+\frac{\partial f}{\partial y}\frac{\partial \cos x}{\partial x}$$
    and see every term if one can be cancelled.
     
  4. Jun 6, 2016 #3

    S.R

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    Ah, thank-you. The first term cancels, right?
     
  5. Jun 6, 2016 #4
    Yes, for the ##\sin y## doesn't depend on ##x.##
     
  6. Jun 6, 2016 #5
    However, forgetting to remind, in the calculation, mind the terms regarding ##f## you have to plug in its pair ##(\sin y,\cos x).##
     
  7. Jun 6, 2016 #6

    S.R

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    Should it be d/dx(siny) as in the single var. case?
     
    Last edited: Jun 6, 2016
  8. Jun 6, 2016 #7
    To be precise, you are right, but using partial derivative doesn't make no sense based on the definition also haha (just for my laziness)
     
  9. Jun 6, 2016 #8

    S.R

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    How would I obtain g_y though? I'm also not sure how to derive the first formula.
     
    Last edited: Jun 6, 2016
  10. Jun 6, 2016 #9
    Well...then maybe you don't understand the chain rule totally...For reference: https://en.wikipedia.org/wiki/Chain_rule
    To make yourself really acquire the information, I suggest getting the full picture of it including the proof.
     
  11. Jun 6, 2016 #10

    S.R

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    The way I learned the chain rule (in the context of multivariable functions) was to draw a dependency diagram. In this case, however, the dependency diagram is not clear.
     
  12. Jun 6, 2016 #11

    Ray Vickson

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    When I learned this stuff we were advised to use notation like ##f_1##, which is "partial derivative of ##f## with respect to the first variable" and ##f_2## for "... with respect to the second variable". That way, when you swap the positions of ##x## and ##y## (as done in this problem) you avoid getting yourself hopelessly confused.
     
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