# Differentiability

• Feb 25th 2009, 04:20 PM
nmatthies1
Differentiability
Using the fact that $\displaystyle x \rightarrow logx$ is differentiable on $\displaystyle (0,\infty)$ with derivative $\displaystyle x \rightarrow \frac {1} {x}$ verify that $\displaystyle arctanh x=\frac {1} {2} log ( \frac {1+x} {1-x})$ is differentiable and compute its derivative. Now I have no problem finding the derivative, however I'm not sure how to show it is differentiable. It shouldn't be too hard but I'm not too acquainted with the method. Thanks!!
• Feb 25th 2009, 06:28 PM
tah
Quote:

Originally Posted by nmatthies1
Using the fact that $\displaystyle x \rightarrow logx$ is differentiable on $\displaystyle (0,\infty)$ with derivative $\displaystyle x \rightarrow \frac {1} {x}$ verify that $\displaystyle arctanh x=\frac {1} {2} log ( \frac {1+x} {1-x})$ is differentiable and compute its derivative. Now I have no problem finding the derivative, however I'm not sure how to show it is differentiable. It shouldn't be too hard but I'm not too acquainted with the method. Thanks!!

Hi, I think you must specify that it's differentiable somewhere. Indeed it's only well defined for $\displaystyle x\in]-1,1[$, so you can use the result on the composition of differentiable functions in that interval.
• Feb 25th 2009, 06:40 PM
nmatthies1
makes sense. thanks!!