STEP III 1990 question 5 solution

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By induction. Case when n=1 is clearly just the product rule:

Assume true for n=k, then differentiate again to get:

Which is the proposition with n=(k+1), therefore by induction it is true for all natural n.

Therefore, the proposition for n=0 reduces to: which is obviously true.

Now assume that the proposition is true for n=k and differentiate:

Which is the proposition for n=k+1. Therefore, by induction, the proposition is true for all non-negative integer n.

Solution by Speleo.