Question

Differentiating a Logarithmic Function In Exercise, find the derivative of the function.

y =In (x^2 - 2)2/3

Answer 1

`y' = (4x)/(3(x^(2) - 2))`

Explanation:

The derivative of
`(f(x))^(n)` is
`f'(x)*n*(f(x))^(n-1)`. The derivative of
`(20x^(2) + 3)^(10)` is
`400x*(20x^(2) + 3)^(9)`, for example.

We also have that the derivative of

`y = ln(f(x))`

Is

`y' = (f'(x))/(f(x))`

In this problem, we have that:

`y = \ln{(x^(2)-2)^{(2)/(3)}}`

So

`f(x) = (x^(2) - 2)^{(2)/(3)}`

`f'(x) = (2*2x)/(3)*(x^(2) - 2)^{-(1)/(3)}`

The derivative of the function is:

`y' = (f'(x))/(f(x))`

`y' = \frac{(2*2x)/(3)*(x^(2) - 2)^{-(1)/(3)}}{(x^(2)-2)^{(2)/(3)}}`

`y' = (4x)/(3(x^(2) - 2))`