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I Derivative When Substituting Variables

  1. Jun 18, 2017 #1

    Drakkith

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    Staff: Mentor

    I'm working through a proof in my differential equations book, but I think I'm hung up on a basic calculus derivative.

    If we have a function ##f(x,y)## and we substitute ##v=\frac{y}{x}## , rearrange to get ##y=vx##, and then take the derivative, supposedly by the product rule we get $$\frac{dy}{dx}=v+x\frac{dv}{dx}$$
    I'm not quite sure how this works since ##v## is a function of both y and x and y itself is a function of x. What's going on here?
     
    Last edited: Jun 19, 2017
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  3. Jun 18, 2017 #2

    cnh1995

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    I think it should be dy/dx=v+x*dv/dx as per the product rule.
     
  4. Jun 19, 2017 #3

    Drakkith

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    I'm sorry, I forgot to add the X in the 2nd term. My mistake.
    I'm still not sure what's going on though. :biggrin:
     
  5. Jun 19, 2017 #4

    Mark44

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    ##v = \frac y x##, so in a sense v is a function of both x and y, but the assumption is that y is a function of x. This means that v is also a function of x alone.

    Starting with the equation y = vx, differentiate both sides with respect to x. This gives you y' = v + v'x, just using the product rule.
     
  6. Jun 19, 2017 #5

    Drakkith

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    Okay. I figured it was something easy I was missing. Thanks guys.
     
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