Question

Differentiate:

give me a moment...

Answer 1

\[y=\frac{ \sqrt[3]{x + \frac{ 2 }{ x }} }{ \sqrt[3]{x-\frac{ 2 }{ x }} }\]

Answer 2

Lets MAYBE make our life a little easier.

\[\frac{ \sqrt[3]{x+\frac{ 2 }{ x }} }{ \sqrt[3]{x-\frac{ 2 }{ x }} }=\sqrt[3]{\frac{ x+\frac{ 2 }{ x } }{ x-\frac{ 2 }{ x } }}=\]
\[\sqrt[3]{\frac{ x^{2}+2 }{ x^{2}-2 }}\]

Answer 3

So chain rule, right? We can consider our outer function to be the cube root and our inner function to be the division inside of the cube root. So since this is chain rule, we take the derivative of each layer (or inner function if you wish) and multiply their results. So we would deal with the cube root portion first using power rule then multiply it by the derivative of whats inside using the quotient rule. Of course, whatever is inside of each layer is not changed. So we would have:
\[\frac{ 1 }{ 3 }(\frac{ x^{2}+2 }{ x^{2}-2 })^{-2/3}\]Thats the first derivative. Now take the derivative of the inner quotient and multiply it by this result.

Answer 4

ahhh!!! thanks!!

Answer 5

yeah, try it out and see how it works out :3

Answer 6

yes it works, i got the correct answer:)

Answer 7

alright, awesome xD