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Thread: Trying to find derivatives for a certain situation

  1. #1
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    Trying to find derivatives for a certain situation

    I wonder if someone could help here. I have the following situation:

    I have the equation:
    g(z) = z - f(z,p,q) = 0

    also z is a function of p and q

    I have already solved this for fixed p and q using Newton-Raphson and hence I have already calculated partial df/dz

    Now I need to calculate partial derivatives dz/dp and dz/dq.

    I can't work out how to do this.
    Last edited by acferrad; Nov 23rd 2016 at 06:58 AM.
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  2. #2
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    Re: Trying to find derivatives for a certain situation

    Actually I think I have it. For partial derivative dz/dp when q is constant:

    dz/dp = -d/dp ( z - f(z,p,q) ) / d/dz ( z - f(z,p,q) ) (d operators are partial)

    so I just need to calculate df/dp at constant z,q in my N-R iteration (and df/dq at constant z,p)

    I think.
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  3. #3
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    Re: Trying to find derivatives for a certain situation

    If z= f(z, p, q) and p and q are also functions of z then, by the chain rule, \frac{\partial z}{\partial p}= \frac{\partial f}{\partial z}\frac{\partial z}{\partial p}+ \frac{\partial f}{\partial p} so that \frac{\partial z}{\partial p}- \frac{\partial f}{\partial z}\frac{\partial z}{\partial u}= \frac{\partial z}{\partial p}\left(1- \frac{\partial f}{\partial z}\right)= \frac{\partial f}{\partial p} and \frac{\partial z}{\partial p}= \frac{\frac{\partial f}{\partial p}}{1- \frac{\partial f}{\partial z}}.

    Similarly for \frac{\partial z}{\partial q}.
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  4. #4
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    Re: Trying to find derivatives for a certain situation

    Thanks HallsofIvy

    One correction to your comment: p and q are not functions of z. rather the reverse: z is a function of p and q.

    p and q are independent variables
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