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Thread: Implicit differentiation

  1. #1
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    Implicit differentiation

    How do i find the answer for these?
    Implicit differentiation-sketch-8-.png
    E.g. for dp/dr = L/F(L)
    But for dL/dr = PF'(L)-r+PF''(L)/PF(L)*F''(L)
    Which is not consistent with the answers in the mark scheme.
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  2. #2
    MHF Contributor
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    Re: Implicit differentiation

    You can solve the first equation algebraically to get F'=\frac{r}{p}. Then differentiate the second equation with respect to r:

    \frac{\partial{p}}{\partial{r}}F+\frac{\partial{F}  }{\partial{r}}p-L-r\frac{\partial{L}}{\partial{r}}-\frac{\partial{B}}{\partial{r}}=0

    \frac{\partial{B}}{\partial{r}}=0 since they are both independent variables. Using the chain rule, since F is a function of L,

    \frac{\partial{F}}{\partial{r}} = F'\frac{\partial{L}}{\partial{r}} = \frac{r}{p}\frac{\partial{L}}{\partial{r}}. So

    \frac{\partial{p}}{\partial{r}}F+\frac{r}{p}\frac{  \partial{L}}{\partial{r}}p-L-r\frac{\partial{L}}{\partial{r}}=0

    The second and fourth terms cancel, and so you get \frac{\partial{p}}{\partial{r}}=\frac{L}{F}.

    I haven't done the other three derivatives, but I would guess that they're similar. If you have the answers already, that usually helps....

    - Hollywood
    Last edited by hollywood; May 17th 2017 at 06:44 PM. Reason: formatting
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