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Math Help - Leibniz rule for differentiating the product, help please!

  1. #1
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    Leibniz rule for differentiating the product, help please!

    If n is a nonnegative integer and functions f and g have nth-order derivatives, show that:
    (fg)^{(n)}=f^{(n)}g+\left(\begin{array}{ll}n\\1\en  d{array}\right)f^{(n-1)}g'+\left(\begin{array}{ll}n\\2\end{array}\right  )f^{(n-2)}g''+...+\left(\begin{array}{ll}n\\n\end{array}\  right)fg^{(n)}
    =\sum_{k=0}^n \left(\begin{array}{ll}n\\k\end{array}\right)f^{(n-k)}g^{(k)}
    Im not well versed in binomial expansion so this is confusing to me
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  2. #2
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    Quote Originally Posted by binkypoo View Post
    If n is a nonnegative integer and functions f and g have nth-order derivatives, show that:
    (fg)^{(n)}=f^{(n)}g+\left(\begin{array}{ll}n\\1\en  d{array}\right)f^{(n-1)}g'+\left(\begin{array}{ll}n\\2\end{array}\right  )f^{(n-2)}g''+...+\left(\begin{array}{ll}n\\n\end{array}\  right)fg^{(n)}
    =\sum_{k=0}^n \left(\begin{array}{ll}n\\k\end{array}\right)f^{(n-k)}g^{(k)}
    Im not well versed in binomial expansion so this is confusing to me

    This is a simple proof by induction: do for n= 1,2 so that you'll get some insight in what's going on here.

    Tonio
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