Question

How do I find the derivative of `f(x)=ln(abs(x))`?

Answer 1

`f'(x)=1/x`

Explanation:

This can be split into a piecewise function.

`f(x)={(ln(x)",","if "x>0),(ln(-x)",","if "x<0):}`

Find the derivative of each part:

`d/dx(ln(x))=1/x`

`d/dx(ln(-x))=1/(-x)*d/dx(-x)=1/x`

Hence,

`f'(x)={(1/x",","if "x>0),(1/x",","if "x<0):}`

This can be simplified, since they're both `1/x`:

`f'(x)=1/x`

Even though `0` wasn't specified in the piecewise function, there is a domain restriction in `1/x` at `x=0` as well.