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Partial of power function

  1. Dec 17, 2013 #1
    Is there a derivation for ∂f(x,y)/∂x given:

    f(x,y): g(x,y)h(x,y)

    e.g. sin(x)(x+2y)
     
  2. jcsd
  3. Dec 17, 2013 #2

    HallsofIvy

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    Yes, of course. Given [itex]f(x)= g(x)^{h(x)}[/itex] we have [itex]ln(f(x))= h(x)ln(g(x))[/itex], then [itex]\frac{1}{f(x)}\frac{df}{dx}= ln(g(x))\frac{dh}{dx}+ \frac{h(x)}{g(x)}\frac{dg}{dx}[/itex]

    So [tex]\frac{df}{dx}= g(x)^{h(x)}\left(ln(g(x))\frac{dh}{dx}+ \frac{h(x)}{g(x)}\frac{dg}{dx}\right)[/tex]

    Of course, the same is true if g and h are functions of x and y and you are taking the derivative with respect to x because you are treating y as a constant.

    (This has nothing to do with differential equations.)
     
  4. Dec 21, 2013 #3
    Thanks

    Ah, I got it, I see what you did! Sometimes when puzzled in an instant with mysterious delight the answer appears. Thanks for the surprise gift - and problem solution.
     
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