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Partial derivative of a single variable function

  1. Feb 26, 2012 #1
    So I don't understand why if you have something like U(x,y) = f(y+2x)

    and you take [tex]\frac{\partial U}{\partial x}
    = \frac{\partial f}{\partial x} [/tex]

    you get [tex]\frac{df}{d(y+2x)} * \frac{d(y+2x)}{dx}[/tex]

    Why does the partial derivative just change to the total derivative for one variable? It seems like you just treat x and y+2x as the same variable? thanks for any help!!
     
  2. jcsd
  3. Feb 26, 2012 #2

    Fredrik

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    Define a function g by g(x,y)=y+2x for all x,y. You have specified that the relationship between U, f and g is given by
    $$U(x,y)=f(g(x,y))=(f\circ g)(x,y)$$ for all x,y. So ##U=f\circ g##, and therefore $$\frac{\partial U}{\partial x}=\frac{\partial (f\circ g)}{\partial x}.$$ Since we're dealing with a composition of functions, we need the chain rule. It tells us that the right-hand side is $$=f'\, \frac{\partial g}{\partial x}.$$ This is often written as $$\frac{\partial f}{\partial g}\frac{\partial g}{\partial x}.$$ The g in the denominator can serve as a reminder that f' is to be evaluated at g(x,y).
     
    Last edited: Feb 26, 2012
  4. Feb 26, 2012 #3
    Thanks! I just wasn't thinking of y+2x as a separate function of x and y.
     
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