Nth derivative and proof by induction

[adwong1]

Hi, I'm having a bit of trouble with finding a general formula for the 'nth derivative' for the function f(x)= x^2/3

So far I've figured out the formula (i think it's right), but I can't figure out how to prove it by induction, since I've never had to prove something like this before.

http://img354.imageshack.us/img354/3484/devoir8awn5.jpg


http://img412.imageshack.us/img412/776/devoir8bbe5.jpg

Thanks in advance. I tried to lay it out as neatly as i possibly could

Alex

 

[Mark44]

I understand what you're trying to say for the powers of x, but your expression
$$f^{(n-1)} (x ^{-1})$$ doesn't mean what you intend it to mean.

You have written $$f^{(n)} (x)$$ recursively. I think that your recursive definition won't help you in your proof by induction, since you won't be able to take the derivative of the right hand side. If you had the right side written nonrecursively, you would be able to take its derivative. In the coefficient of the derivative expression, you have a fraction, with the numerator being a product of 2(-1)(-4) and so on, and the denominator being 3 raised to the same power as there are factors in the numerator. The exponent on x will be 2/3 - that same number. Maybe you can work with these ideas.