Question

Differentiating a Logarithmic Function in Exercise, find the derivative of the function. See Examples 1, 2, 3, and 4.

y = ln(x + 1/x - 1)^1/2

Answer 1

`(dy)/(dx)=- (1)/(\left(x - 1\right) \left(x + 1\right))`

Explanation:

Given:

`y = \ln{\left((x + 1)/(x - 1)\right)^{(1)/(2)}`

using the properties of log we can take the power 1/2 and multiply it.

`y = (1)/(2)\ln{\left((x + 1)/(x - 1)\right)`

now we can differentiate:

`(dy)/(dx) = (1)/(2)(1)/(\left((x + 1)/(x - 1)\right))\left((d)/(dx)\left((x+1)/(x-1)\right)\right)`

`(dy)/(dx) = (1)/(2)\left((x - 1)/(x + 1)\right)\left(- (2)/(\left(x - 1\right)^(2))\right)`

`(dy)/(dx)=- (1)/(\left(x - 1\right) \left(x + 1\right))`

this is our answer!