Answer:
[tex]y' = \frac{4x}{3(x^{2} - 2)}[/tex]
Step-by-step explanation:
The derivative of [tex](f(x))^{n}[/tex] is [tex]f'(x)*n*(f(x))^{n-1}[/tex]. The derivative of [tex](20x^{2} + 3)^{10}[/tex] is [tex]400x*(20x^{2} + 3)^{9}[/tex], for example.
We also have that the derivative of
[tex]y = \ln{f(x)}[/tex]
Is
[tex]y' = \frac{f'(x)}{f(x)}[/tex]
In this problem, we have that:
[tex]y = \ln{(x^{2}-2)^{\frac{2}{3}}}[/tex]
So
[tex]f(x) = (x^{2} - 2)^{\frac{2}{3}}[/tex]
[tex]f'(x) = \frac{2*2x}{3}*(x^{2} - 2)^{-\frac{1}{3}}[/tex]
The derivative of the function is:
[tex]y' = \frac{f'(x)}{f(x)}[/tex]
[tex]y' = \frac{\frac{2*2x}{3}*(x^{2} - 2)^{-\frac{1}{3}}}{(x^{2}-2)^{\frac{2}{3}}}[/tex]
[tex]y' = \frac{4x}{3(x^{2} - 2)}[/tex]
