Differential of Arctanh x

To find  
\[\frac{d}{dx}(coth^{-1}x)\]
 ;et  
\[y=coth^{-1}x\]
  then  
\[cothy=x\]
.
Differentiating implicitly gives  
\[- cosech^2 x \frac{dy}{dx}=1 \rightarrow \frac{dy}{dx}=- \frac{1}{cosech^2 y}\]
.
To express  
\[\frac{dy}{dx}\]
  in terms of  
\[x\]
  use  
\[cothy=x,\; cosech^2y =coth^2y-1=x^2-1\]
.
Then  
\[\frac{dy}{dx}=- \frac{1}{x^2-1}\]
.

Add comment

Security code
Refresh