Question

What is the derivative of `f(x)=-ln(1/x)+lnx`?

Answer 1

`f'(x)=2/x`

Explanation:

We could jump right in and apply the Chain Rule, but it is worthwhile to apply the very convenient properties of logarithms. It will avoid a lot of messy work and eliminate room for trivial errors.

`ln(1/x)=ln(1)-ln(x)`

`ln(1)=0,` so

`ln(1/x)=-lnx` and we end up with

`f(x)=-(-lnx)+lnx=lnx+lnx=2lnx`

Now, recalling that `d/dxlnx=1/x,`

`f'(x)=2/x`