What Is the Derivative of Arctanh(x/2)?

[ultima9999]

Yeah, I was working through this problem and it differs from the answer that my friend got.

Using implicit differentiation, find the derivative of \mbox{arc}\tanh \frac{x}{2} and state the domain for which the derivative applies

\begin{align*}<br /> y = \arctanh \frac{x}{2} \\<br /> \Leftrightarrow x = 2 \tanh y<br /> \end{align*}

\frac{d}{dx}x = \frac{d}{dx}2 \tanh y
\Rightarrow 1 = 2\ \mbox{sech}^2 y \cdot \frac{dy}{dx}
\Rightarrow \frac{dy}{dx} = \frac{1}{2\ \mbox{sech}^2 y}
\Rightarrow \frac{dy}{dx} = \frac{1}{2 - 2 \tanh^2 y}
\Rightarrow \frac{dy}{dx} = \frac{1}{2 - 2 \frac{x^2}{4}}
\Rightarrow \frac{dy}{dx} = \frac{1}{2 - \frac{x^2}{2}}

 

[0rthodontist]

The derivative of what now? d(x/2)/dx = 1/2. I assume you mean tanh^-1(x/2).

Yeah, you're right.