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Math Help - Derivative formular for F(x)=f(x)g(x)

  1. #1
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    Derivative formular for F(x)=f(x)g(x)

    F(x) = f(x)g(x)
    F^1 = f^1 g + g^1  f
    F^2 = f^2 g + 2 f^1g^1+g^2f
    F^3=f^3g+3f^2g^1+3f^1g^2+fg^3
    F^4=f^4g+4f^3g^1+6f^2g^2+4f^1g^3+fg^4

    could you help to find the fomular for

    F^n= ????
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  2. #2
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    Quote Originally Posted by thangbe View Post
    F(x) = f(x)g(x)
    F^1 = f^1 g + g^1 f
    F^2 = f^2 g + 2 f^1g^1+g^2f
    F^3=f^3g+3f^2g^1+3f^1g^2+fg^3
    F^4=f^4g+4f^3g^1+6f^2g^2+4f^1g^3+fg^4

    could you help to find the fomular for

    F^n= ????
    http://en.wikipedia.org/wiki/Leibniz...d_product_rule)
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  3. #3
    MHF Contributor matheagle's Avatar
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    It's just the binomial theorem with derivatives instead of powers.
    So the coefficients are  {n \choose k}={n! \over k!(n-k)!}
    and f^{(0)}=f.
    Last edited by matheagle; March 19th 2009 at 08:09 PM.
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  4. #4
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    Quote Originally Posted by matheagle View Post
    It's just the binomial theorem with derivatives instead of powers.
    So the coefficients are {n \choose k}
    and f^{(0)}=f.

    thank you
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