Sign of partial derivative from implicit function

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I have a F.O.C that implicitly defines a function



$x^*(b_R, pi)$



Given that I cannot solve for $x^*$ and that I am interested in $fracdx^*db_R$ and $fracdx^*dpi$ I implement implicit differentiation with respect to the variable of interest.



I then obtain



$fracdx^*db_R = some function of x, pi,b_R$.



by definition all of $x, pi, b_R in (0,1)$. I then plot the derivative with respect to say b_R, with $x$ on the x-axis.



However I still cannot figure out when this derivative is positive or negative as I vary $pi$ or $b_R$ (which are predetermined).



Here is the Desmos link, the derivates are the first and 9th functions:



https://www.desmos.com/calculator/rrcptzleok




derivatives optimization implicit-differentiation




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    I have a F.O.C that implicitly defines a function



    $x^*(b_R, pi)$



    Given that I cannot solve for $x^*$ and that I am interested in $fracdx^*db_R$ and $fracdx^*dpi$ I implement implicit differentiation with respect to the variable of interest.



    I then obtain



    $fracdx^*db_R = some function of x, pi,b_R$.



    by definition all of $x, pi, b_R in (0,1)$. I then plot the derivative with respect to say b_R, with $x$ on the x-axis.



    However I still cannot figure out when this derivative is positive or negative as I vary $pi$ or $b_R$ (which are predetermined).



    Here is the Desmos link, the derivates are the first and 9th functions:



    https://www.desmos.com/calculator/rrcptzleok




    derivatives optimization implicit-differentiation




    share|cite|improve this question





















      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      I have a F.O.C that implicitly defines a function



      $x^*(b_R, pi)$



      Given that I cannot solve for $x^*$ and that I am interested in $fracdx^*db_R$ and $fracdx^*dpi$ I implement implicit differentiation with respect to the variable of interest.



      I then obtain



      $fracdx^*db_R = some function of x, pi,b_R$.



      by definition all of $x, pi, b_R in (0,1)$. I then plot the derivative with respect to say b_R, with $x$ on the x-axis.



      However I still cannot figure out when this derivative is positive or negative as I vary $pi$ or $b_R$ (which are predetermined).



      Here is the Desmos link, the derivates are the first and 9th functions:



      https://www.desmos.com/calculator/rrcptzleok




      derivatives optimization implicit-differentiation




      share|cite|improve this question











      I have a F.O.C that implicitly defines a function



      $x^*(b_R, pi)$



      Given that I cannot solve for $x^*$ and that I am interested in $fracdx^*db_R$ and $fracdx^*dpi$ I implement implicit differentiation with respect to the variable of interest.



      I then obtain



      $fracdx^*db_R = some function of x, pi,b_R$.



      by definition all of $x, pi, b_R in (0,1)$. I then plot the derivative with respect to say b_R, with $x$ on the x-axis.



      However I still cannot figure out when this derivative is positive or negative as I vary $pi$ or $b_R$ (which are predetermined).



      Here is the Desmos link, the derivates are the first and 9th functions:



      https://www.desmos.com/calculator/rrcptzleok





      derivatives optimization implicit-differentiation





      share|cite|improve this question










      share|cite|improve this question




      share|cite|improve this question









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