Date: Tue, 11 Mar 1997 23:08:12 -0800 (PST)
From: Shuling
Subject: Mathematical Induction and the derivative

Please help answer this question:

"Obtain a formula for the nth derivative of the product of two functions, and prove the formula by induction on n."

Any educated tries are appreciated.

Hi Shuling

If you take two functions, f and g, and find the first, second and third derivative of fg you should see the binomial coeficients and come up with a hypothesis that

Here designates the jth derivative of f and .

It is clear that this is valid if n=1 and hence for the induction step calculate

But so we get

Since we can write this

which is the hypothezised expression for the (n+1)st derivative of fg. Thus

for all positive integers n.


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