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How did v come out of this derivative?

  1. Aug 5, 2012 #1
    How did "v" come out of this derivative?

    1. The problem statement, all variables and given/known data


    ffa.jpg



    3. The attempt at a solution

    I basically understood how to attack the problem and my answer was very close to the key. The only thing bothering me is that v on the top. I cannot see how that could come up anywhere.

    I know that the "r" is r = r(t), so that I am only taking the derivative of $$\ln(1 + w/r)$$. I don't see where the v actually is
     
  2. jcsd
  3. Aug 5, 2012 #2

    TSny

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    Re: How did "v" come out of this derivative?

    The flux is expressed as a function of r. You're taking the derivative with respect to t. So, recall the chain rule of calculus: first take the derivative with respect to r and then multiply by the derivative of r with respect to t.
     
  4. Aug 6, 2012 #3
    Re: How did "v" come out of this derivative?

    Nope, implicit differentiation. I had forgotten about that
     
  5. Aug 6, 2012 #4

    TSny

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    Re: How did "v" come out of this derivative?


    Here's the chain rule: If [itex]y=f[/itex] and [itex]u = g[x][/itex] are differentiable functions, then [itex]\frac{dy}{dx}[/itex]=[itex]\frac{dy}{du}[/itex][itex]\cdot[/itex][itex]\frac{du}{dx}[/itex]

    You have the flux expressed as a function of r: [itex]\Phi[r][/itex] and r is some function of time: [itex]r[t][/itex].

    So, the chain rule says [itex]\frac{d\Phi}{dt}[/itex] = [itex]\frac{d\Phi}{dr}[/itex][itex]\cdot[/itex][itex]\frac{dr}{dt}[/itex]

    This get's you the answer and it shows why the speed v appears.

    How do you get the answer using implicit differentiation?
     
  6. Aug 6, 2012 #5
    Re: How did "v" come out of this derivative?



    No you are right, I forget the $\phi$ on the LHS. I was thinking something like

    Something = r as a function of t ==> differentiate both sides wrt t ==> implicitly differentiate RHS

    http://www.wolframalpha.com/input/?i=D[ln%281+%2B+w%2Fr%28t%29%29%2Ct]
     
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