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Total derivative with a constraint

  1. Sep 3, 2013 #1
    Hi there,

    I have what I suspect is a straightforward question.

    I wish to take the total derivative of the following function:

    [itex]W(q,x) = q \cdot u(x) + c(q,x)[/itex]

    Subject to the constraint: [itex]\frac{q}{x}[/itex]=[itex]\bar{m}[/itex], where [itex]\bar{m}[/itex] is some constant > 0, and c(q,x) is additively separable.

    Without the constraint the total derivative is simply:

    [itex] dW(q,x) = u(x) dq + q \cdot u_{x} dx + c_{q}(q,x) dq + c_{x}(q,x) dx [/itex]

    My question is: How do I incorporate the constraint?

    Thanks for any help!

  2. jcsd
  3. Sep 3, 2013 #2


    User Avatar
    Science Advisor

    The constraint seems to be q = xm, so dq = mdx.
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