Quote:
Originally Posted by ThePerfectHacker  2)Let  and  be functions which you can differenciate. Note that ![[f(x)g(x)]' = f'(x)g(x)+f(x)g'(x)](http://www.mathhelpforum.com/math-help/latex2/img/ea54776467b9e45791138a031d6d92f7-1.gif "[f(x)g(x)]' = f'(x)g(x)+f(x)g'(x)") . Can you find a formula for ![[f(x)g(x)]^{(n)}](http://www.mathhelpforum.com/math-help/latex2/img/e81fb9be7ed24bed346737eb78216192-1.gif "[f(x)g(x)]^{(n)}") where by  means to preform differenciation  times repeatedly. |
After a little experimenting:
![[f(x)g(x)]'=f'(x)g(x) + f(x)g'(x)](http://www.mathhelpforum.com/math-help/latex2/img/aa3cec18b4af42e9424ff749b93ec679-1.gif "[f(x)g(x)]'=f'(x)g(x) + f(x)g'(x)")
![[f(x)g(x)]''=f''(x)g(x)+2f'(x)g'(x)+f(x)g''(x)](http://www.mathhelpforum.com/math-help/latex2/img/0660ca97d29cf98c36b81c5296a99112-1.gif "[f(x)g(x)]''=f''(x)g(x)+2f'(x)g'(x)+f(x)g''(x)")
![[f(x)g(x)]''' = f'''(x)g(x)+3f''(x)g'(x)+3f'(x)g''(x)+f(x)g'''(x)](http://www.mathhelpforum.com/math-help/latex2/img/599d2f725675a07a39e3cd9b65d0217d-1.gif "[f(x)g(x)]''' = f'''(x)g(x)+3f''(x)g'(x)+3f'(x)g''(x)+f(x)g'''(x)")
This reminds me of the binomial theorem:
So perhaps
![[f(x)g(x)]^{(n)} = \sum_{k=0}^n {n\choose k} f^{n-k}(x) g^{k}(x)](http://www.mathhelpforum.com/math-help/latex2/img/c5492a7ccb489ebb136dce53602f59ba-1.gif "[f(x)g(x)]^{(n)} = \sum_{k=0}^n {n\choose k} f^{n-k}(x) g^{k}(x)")
Where
denotes the nth derivative of
Nice problem
P.S Is there any way to hide text?