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.

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.

Cheers
Penny