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Can h(x)=(cos x)^x be written as a composition of two functions f and

  1. Jul 14, 2010 #1
    Can h(x)=(cos x)^x be written as a composition of two functions f and g where f(x)=x^n and g(x)=cosx ? where h=fog

    REASON FOR ASKING: I am wondering this in connect with a differentiation I was having trouble with (but can now solve thanks to this forum). I mistakenly thought that I could apply the chain rule for composition of functions. Seems it doesn't apply. (https://www.physicsforums.com/showthread.php?p=2796762#post2796762)
     
  2. jcsd
  3. Jul 14, 2010 #2
    Re: Composition

    You need to be careful with notation. In your example
    [tex]h(x)=f\circ g(x)=(\cos(x))^n[/tex]
    where you have a different variable in the exponent. So all rules you know are valid without exception, but you have to get the notation right.

    You could also try
    f(x)=a^x or f(x,n)=x^n
    but you'll notice that at some point the expression won't match what you have in your rules.
     
  4. Jul 14, 2010 #3

    Gib Z

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    Re: Composition

    It can't, because in h(x) the exponent is x, the variable, while in f(x) the exponent is a constant. This makes an important difference when you are differentiating because the standard "power law" only applies when the exponent is constant.
     
  5. Jul 14, 2010 #4

    HallsofIvy

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    Re: Composition

    Specifically, the derivative of xa, with a constant, is axa-1 while the derivative of ax is (ln(a))ax.
     
  6. Jul 15, 2010 #5
    Re: Composition

    Thanks. I can see my mistake --- there's no way to define f(x) to satisfy the requirements.
     
  7. Jul 15, 2010 #6

    Gib Z

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    Re: Composition

    There is. Remember that [tex]f(x) = e^{\ln f(x)}[/tex]
     
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