Question

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

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

Answer 1

Answer:
`(2x^2-1)/(x(x^2-1))`

Explanation:

The given function :
`y=\ln(x(x^2 - 1)^{(1)/(2)})`

`\Rightarrow\ y=\ln x+\ln (x^2-1)^{(1)/(2)}` [
`\because \ln(ab)=\ln a +\ln b`]

`\Rightarrow y=\ln x+(1)/(2)\ln (x^2-1)}` [
`\because \ln(a)^n=n\ln a`]

Now , Differentiate both sides with respect to x , we will get

`(dy)/(dx)=(1)/(x)+(1)/(2)((1)/(x^2-1))(d)/(dx)(x^2-1)` (By Chain rule)

[Note :
`(d)/(dx)(\ln x)=(1)/(x)`]

`(1)/(x)+(1)/(2)((1)/(x^2-1))(2x-0)`

[
`\because (d)/(dx)(x^n)=nx^(n-1)`]

`=(1)/(x)+(1)/(2)((1)/(x^2-1))(2x) = (1)/(x)+(x)/(x^2-1)\\\\\\=((x^2-1)+(x^2))/(x(x^2-1))\\\\\\=(2x^2-1)/(x(x^2-1))`

Hence, the derivative of the given function is
`(2x^2-1)/(x(x^2-1))` .