Differentiation - cant remember

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Homework Statement



for the question and attempt see the attachment. I've nearly done this but from my ti-89 I know that i am missing a 2x from the top line and i can't find where its gone. been a while since I've done this and i think that its just a simple mistake... ps it does want the double differential but once i get the first diff right i should be able to finish it off... thanks for any help.

Homework Equations





The Attempt at a Solution

 

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pat666 said:

Homework Statement



for the question and attempt see the attachment. I've nearly done this but from my ti-89 I know that i am missing a 2x from the top line and i can't find where its gone. been a while since I've done this and i think that its just a simple mistake... ps it does want the double differential but once i get the first diff right i should be able to finish it off... thanks for any help.

Homework Equations





The Attempt at a Solution

The 2x is missing because you forgot that you need to use the chain rule.
 
so i did- thanks for that.
 
There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...

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